
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 $a= \int_{0}^{\frac{\pi}{2}} \left( \sin \frac{x}{2} \right)^2 - \frac{1}{2} \,dx$, determine the coefficient of the term in $x$ in the expansion of $\left(ax+ \frac{1}{2ax}\right)^9$.

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