
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
We consider a simple model for balanced parenthesis checking. Let  $\mathcal R=\{\texttt{(())}\rightarrow \texttt{A},\texttt{(A)}\rightarrow\texttt{A},\texttt{AA}\rightarrow\texttt{A}\}$  be a set of rules for phrase reduction. Ideally, any given phrase is balanced if and only if the model is able to reduce the phrase to  $\texttt{A}$  by some arbitrary sequence of rule applications. For example, to show  $\texttt{((()))}$  is balanced we can perform the following sequence of reductions.
\[\texttt{((()))}\rightarrow\texttt{(A)}\rightarrow\texttt{A}\qquad \checkmark\]
Unfortunately, the above set of rules  $\mathcal R$  is not complete, since there exist parenthetical phrases which are balanced but which are not balanced according to  $\mathcal R$ .  Determine the number of such phrases of length  $14$ .

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