
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 the binomial expansion of $\left( x+\frac{1}{2ax} \right)^{9}$ where the coefficient of $x^{3}$ is $-\frac{21}{2}$, find the value of $\int\_{1}^{e}{\frac{a}{x}dx}+\int\_{0}^{1}{\sqrt{1-{{t}^{2}}}}dt$

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