
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
If the maximum value of the function $f(x)= \frac{1+\cos 2x}{4\sin (\frac{\pi}{2}+x)}-a\sin \frac{x}{2}\cos (\pi-\frac{x}{2})$ is $1$, determine the value of the constant $a$.

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