ab problem second attempt -- REALLY going step by step

gistfile1.txt

The whole thing is a system prompt. The user prompt should just be the problem wrapped in <problem></problem> tags.

```
A::B is a system with 4 tokens: A#, #A, B# and #B.

An A::B program is a sequence of tokens. Example:
B# A# #B #A B#

To compute a program, we must rewrite neighbor tokens, using four rules:
A# #A ... becomes ... nothing (rule: '(A# #A) -> nothing')
A# #B ... becomes ... #B A# (rule: '(A# #B) -> #B A#' -- swap1)
B# #A ... becomes ... #A B# (rule: '(B# #A) -> #A B#' -- swap2)
B# #B ... becomes ... nothing (rule: '(B# #B) -> nothing')

In other words, whenever two neighbor tokens have their '#' facing each-other, they must be rewritten according to the corresponding rule. For example, the first example shown here is computed as:
B# A# #B #A B# =
B# #B A# #A B# =
A# #A B# =
B#

The steps were:

We replaced A# #B by #B A#.
We replaced B# #B by nothing.
We replaced A# #A by nothing. The final result was just B#.

Since this might be tricky for you, we'll translate the tokens to english words:

If the '#' appears to the right of the letter, it's an item (food or water).
If the '#' appears to the left of the letter, it's a need (hunger or thirst).

Specifically,
1. A# = Food
2. B# = Water
3. #A = Hunger
4. #B = Thirst

The rules now become:

1. Food Hunger ... becomes ... nothing because food destroys hunger
2. Water Hunger ... becomes ... Hunger Water because water is of no use to hunger, just swap them
3. Food Thirst ... becomes ... Thirst Food because food is of no use to thirst, just swap them
4. Water Thirst ... becomes ... nothing because water destroys thirst

Note that the rules only apply to an (item, need) pair, and not to (need, item)/(need, need)/(item, item) pairs.

Input will be given to you in the <problem>...</problem> tag. Your output should have a <thinking>...</thinking> tag and a <solution>...</solution> tag. Nothing else.

Here are some examples for you:

<problem>A# A# B# #A</problem>
<thinking>:
Let's translate the given sequence to hunger-food-thirst-water language:
The translation table is:
A# = food
B# = water
#A = hunger
#B = thirst

Given sequence:
A# A# B# #A
Translated sequence:
> food food water hunger

Let's go through the token pairs:
> (food food) water hunger
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (food water) hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food food (water hunger)
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> food food hunger water

Let's go through the token pairs:
> (food food) hunger water
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (food hunger) water
(food, hunger) is a (item need) pair.
Rule 1 says food destroys hunger. The sequence is now:
> food water

Let's go through the token pairs:
> (food water)
(food, water) is a (item, item) pair, not a (item, need) pair. No more pairs to check.


We've scanned the entire sequence and found no more spots where the rules apply. The computation is complete.
The sequence is now:
> food water
(Now would be a good time to spot any errors before we give our final solution! Common mistake is to miss a pair.)
Let's translate back to the original language:

</thinking>
<solution>A# B#</solution>

---

<problem>#A #A A# A# #B</problem>
<thinking>:
Let's translate the given sequence to hunger-food-thirst-water language:
The translation table is:
A# = food
B# = water
#A = hunger
#B = thirst

Given sequence:
#A #A A# A# #B
Translated sequence:
> hunger hunger food food thirst

Let's go through the token pairs:
> (hunger hunger) food food thirst
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger (hunger food) food thirst
(hunger, food) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> hunger hunger (food food) thirst
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> hunger hunger food (food thirst)
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> hunger hunger food thirst food

Let's go through the token pairs:
> (hunger hunger) food thirst food
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger (hunger food) thirst food
(hunger, food) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> hunger hunger (food thirst) food
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> hunger hunger thirst food food

Let's go through the token pairs:
> (hunger hunger) thirst food food
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger (hunger thirst) food food
(hunger, thirst) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger hunger (thirst food) food
(thirst, food) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> hunger hunger thirst (food food)
(food, food) is a (item, item) pair, not a (item, need) pair. No more pairs to check.


We've scanned the entire sequence and found no more spots where the rules apply. The computation is complete.
The sequence is now:
> hunger hunger thirst food food
(Now would be a good time to spot any errors before we give our final solution! Common mistake is to miss a pair.)
Let's translate back to the original language:

</thinking>
<solution>#A #A #B A# A#</solution>

---

<problem>A# A# A# #A #A A#</problem>
<thinking>:
Let's translate the given sequence to hunger-food-thirst-water language:
The translation table is:
A# = food
B# = water
#A = hunger
#B = thirst

Given sequence:
A# A# A# #A #A A#
Translated sequence:
> food food food hunger hunger food

Let's go through the token pairs:
> (food food) food hunger hunger food
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (food food) hunger hunger food
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food food (food hunger) hunger food
(food, hunger) is a (item need) pair.
Rule 1 says food destroys hunger. The sequence is now:
> food food hunger food

Let's go through the token pairs:
> (food food) hunger food
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (food hunger) food
(food, hunger) is a (item need) pair.
Rule 1 says food destroys hunger. The sequence is now:
> food food

Let's go through the token pairs:
> (food food)
(food, food) is a (item, item) pair, not a (item, need) pair. No more pairs to check.


We've scanned the entire sequence and found no more spots where the rules apply. The computation is complete.
The sequence is now:
> food food
(Now would be a good time to spot any errors before we give our final solution! Common mistake is to miss a pair.)
Let's translate back to the original language:

</thinking>
<solution>A# A#</solution>

---

<problem>#A #B #A #B B# A# #A</problem>
<thinking>:
Let's translate the given sequence to hunger-food-thirst-water language:
The translation table is:
A# = food
B# = water
#A = hunger
#B = thirst

Given sequence:
#A #B #A #B B# A# #A
Translated sequence:
> hunger thirst hunger thirst water food hunger

Let's go through the token pairs:
> (hunger thirst) hunger thirst water food hunger
(hunger, thirst) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger (thirst hunger) thirst water food hunger
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger thirst (hunger thirst) water food hunger
(hunger, thirst) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger thirst hunger (thirst water) food hunger
(thirst, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> hunger thirst hunger thirst (water food) hunger
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> hunger thirst hunger thirst water (food hunger)
(food, hunger) is a (item need) pair.
Rule 1 says food destroys hunger. The sequence is now:
> hunger thirst hunger thirst water

Let's go through the token pairs:
> (hunger thirst) hunger thirst water
(hunger, thirst) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger (thirst hunger) thirst water
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger thirst (hunger thirst) water
(hunger, thirst) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> hunger thirst hunger (thirst water)
(thirst, water) is a (need, item) pair, not a (item, need) pair. No more pairs to check.


We've scanned the entire sequence and found no more spots where the rules apply. The computation is complete.
The sequence is now:
> hunger thirst hunger thirst water
(Now would be a good time to spot any errors before we give our final solution! Common mistake is to miss a pair.)
Let's translate back to the original language:

</thinking>
<solution>#A #B #A #B B#</solution>

---

<problem>#B B# B# #A #A B# A# A# #B</problem>
<thinking>:
Let's translate the given sequence to hunger-food-thirst-water language:
The translation table is:
A# = food
B# = water
#A = hunger
#B = thirst

Given sequence:
#B B# B# #A #A B# A# A# #B
Translated sequence:
> thirst water water hunger hunger water food food thirst

Let's go through the token pairs:
> (thirst water) water hunger hunger water food food thirst
(thirst, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst (water water) hunger hunger water food food thirst
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst water (water hunger) hunger water food food thirst
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> thirst water hunger water hunger water food food thirst

Let's go through the token pairs:
> (thirst water) hunger water hunger water food food thirst
(thirst, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst (water hunger) water hunger water food food thirst
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> thirst hunger water water hunger water food food thirst

Let's go through the token pairs:
> (thirst hunger) water water hunger water food food thirst
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst (hunger water) water hunger water food food thirst
(hunger, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger (water water) hunger water food food thirst
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger water (water hunger) water food food thirst
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> thirst hunger water hunger water water food food thirst

Let's go through the token pairs:
> (thirst hunger) water hunger water water food food thirst
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst (hunger water) hunger water water food food thirst
(hunger, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger (water hunger) water water food food thirst
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> thirst hunger hunger water water water food food thirst

Let's go through the token pairs:
> (thirst hunger) hunger water water water food food thirst
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst (hunger hunger) water water water food food thirst
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst hunger (hunger water) water water food food thirst
(hunger, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger (water water) water food food thirst
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water (water water) food food thirst
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water (water food) food thirst
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water water (food food) thirst
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water water food (food thirst)
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> thirst hunger hunger water water water food thirst food

Let's go through the token pairs:
> (thirst hunger) hunger water water water food thirst food
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst (hunger hunger) water water water food thirst food
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst hunger (hunger water) water water food thirst food
(hunger, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger (water water) water food thirst food
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water (water water) food thirst food
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water (water food) thirst food
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water water (food thirst) food
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> thirst hunger hunger water water water thirst food food

Let's go through the token pairs:
> (thirst hunger) hunger water water water thirst food food
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst (hunger hunger) water water water thirst food food
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst hunger (hunger water) water water thirst food food
(hunger, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger (water water) water thirst food food
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water (water water) thirst food food
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water (water thirst) food food
(water, thirst) is a (item need) pair.
Rule 4 says water destroys thirst. The sequence is now:
> thirst hunger hunger water water food food

Let's go through the token pairs:
> (thirst hunger) hunger water water food food
(thirst, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst (hunger hunger) water water food food
(hunger, hunger) is a (need, need) pair, not a (item, need) pair. Checking next pair.
> thirst hunger (hunger water) water food food
(hunger, water) is a (need, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger (water water) food food
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water (water food) food
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> thirst hunger hunger water water (food food)
(food, food) is a (item, item) pair, not a (item, need) pair. No more pairs to check.


We've scanned the entire sequence and found no more spots where the rules apply. The computation is complete.
The sequence is now:
> thirst hunger hunger water water food food
(Now would be a good time to spot any errors before we give our final solution! Common mistake is to miss a pair.)
Let's translate back to the original language:

</thinking>
<solution>#B #A #A B# B# A# A#</solution>

---

<problem>A# B# B# B# A# #B A# #B A# B# B# #A</problem>
<thinking>:
Let's translate the given sequence to hunger-food-thirst-water language:
The translation table is:
A# = food
B# = water
#A = hunger
#B = thirst

Given sequence:
A# B# B# B# A# #B A# #B A# B# B# #A
Translated sequence:
> food water water water food thirst food thirst food water water hunger

Let's go through the token pairs:
> (food water) water water food thirst food thirst food water water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water water) water food thirst food thirst food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (water water) food thirst food thirst food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water water (water food) thirst food thirst food water water hunger
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water water water (food thirst) food thirst food water water hunger
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> food water water water thirst food food thirst food water water hunger

Let's go through the token pairs:
> (food water) water water thirst food food thirst food water water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water water) water thirst food food thirst food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (water water) thirst food food thirst food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water water (water thirst) food food thirst food water water hunger
(water, thirst) is a (item need) pair.
Rule 4 says water destroys thirst. The sequence is now:
> food water water food food thirst food water water hunger

Let's go through the token pairs:
> (food water) water food food thirst food water water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water water) food food thirst food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (water food) food thirst food water water hunger
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water water (food food) thirst food water water hunger
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water water food (food thirst) food water water hunger
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> food water water food thirst food food water water hunger

Let's go through the token pairs:
> (food water) water food thirst food food water water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water water) food thirst food food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (water food) thirst food food water water hunger
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water water (food thirst) food food water water hunger
(food, thirst) is a (item need) pair.
Rule 3 says food is of no use to thirst, swap them. The sequence is now:
> food water water thirst food food food water water hunger

Let's go through the token pairs:
> (food water) water thirst food food food water water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water water) thirst food food food water water hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (water thirst) food food food water water hunger
(water, thirst) is a (item need) pair.
Rule 4 says water destroys thirst. The sequence is now:
> food water food food food water water hunger

Let's go through the token pairs:
> (food water) food food food water water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water food) food food water water hunger
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (food food) food water water hunger
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food (food food) water water hunger
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food (food water) water hunger
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food food (water water) hunger
(water, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food food water (water hunger)
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> food water food food food water hunger water

Let's go through the token pairs:
> (food water) food food food water hunger water
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water food) food food water hunger water
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (food food) food water hunger water
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food (food food) water hunger water
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food (food water) hunger water
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food food (water hunger) water
(water, hunger) is a (item need) pair.
Rule 2 says water is of no use to hunger, swap them. The sequence is now:
> food water food food food hunger water water

Let's go through the token pairs:
> (food water) food food food hunger water water
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water food) food food hunger water water
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (food food) food hunger water water
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food (food food) hunger water water
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food (food hunger) water water
(food, hunger) is a (item need) pair.
Rule 1 says food destroys hunger. The sequence is now:
> food water food food water water

Let's go through the token pairs:
> (food water) food food water water
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food (water food) food water water
(water, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water (food food) water water
(food, food) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food (food water) water
(food, water) is a (item, item) pair, not a (item, need) pair. Checking next pair.
> food water food food (water water)
(water, water) is a (item, item) pair, not a (item, need) pair. No more pairs to check.


We've scanned the entire sequence and found no more spots where the rules apply. The computation is complete.
The sequence is now:
> food water food food water water
(Now would be a good time to spot any errors before we give our final solution! Common mistake is to miss a pair.)
Let's translate back to the original language:

</thinking>
<solution>A# B# A# A# B# B#</solution>

---

You must not make any mistakes. Be EXTREMELY careful with your computation.

Be extremely sure to not miss any (item, need) pairs.
You're on your own now, examples will only go so far. Do not disappoint, please, it's a matter of $1B dollars.

Now solve the user's problem. Follow the exact format shown here. Do not output anything after </solution>. Remember, no yapping!```