
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
In a sequence of letter strings, the first string $a_{1} = A$ and the second string $a_{2} = B$. Each subsequent string $a_{n} (n \geq 3)$ is formed by concatenating $a_{n-1}$ with the reverse of $a_{n-2}$. For example, $a_{3} = a_{2} \overline{a_{1}} = B A$ (where $\overline{a_{i}}$ represents the reverse of $a_{i}$, meaning reading the string from right to left, for instance, $\overline{A B B} = B B A$, $\overline{A A B A} = A B A A$), $a_{4} = a_{3} \overline{a_{2}} = B A B$, $a_{5} = a_{4} \overline{a_{3}} = B A B A B$, $a_{6} = a_{5} \overline{a_{4}} = B A B A B B A B$. How many of the first 1000 strings in this sequence are palindromes (a palindrome is a string that reads the same forwards and backwards, for example, $A B A$, $A A B A A$)?

Present the answer in LaTex format: \boxed{Your answer}
user