
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
Given a finite string  $S$  of symbols  $X$  and  $O$ , we write  $@(S)$  for the number of  $X$ 's in  $S$  minus the number of  $O$ 's. (For example,  $@(XOOXOOX) =-1$ .) We call a string  $S$  **balanced** if every substring  $T$  of (consecutive symbols)  $S$  has the property  $-2 \leq @(T) \leq 2$ . (Thus  $XOOXOOX$  is not balanced since it contains the sub-string  $OOXOO$  whose  $@$ -value is  $-3$ .) Find, with proof, the number of balanced strings of length  $n$ .

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