
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 that $a = \int_{0}^{\frac{\pi}{2}} \left( \frac{1}{2} - \sin^2 \frac{x}{2} \right) dx$, find the coefficient of the term in $x$ when the expression $\left( ax + \frac{1}{2ax} \right)^9$ is expanded.

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