
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
The function $f(x)=A\sin \omega x (\omega > 0)$, for any $x$, satisfies $f\left( x- \frac{1}{2} \right)=f\left( x+ \frac{1}{2} \right)$, and $f\left( -\frac{1}{4} \right)=-a$. Determine the value of $f\left( \frac{9}{4} \right)$.

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