
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
For  $a>1$ , let  $f(x)=\int_{x}^{ax} \sqrt{1-t^2} \,dt\ \left(0\leq x\leq \frac{1}{a}\right).$ (1) Evaluate  $f\left(\frac 1{a\right)}+\int_0^{\frac{1}{a}} \sqrt{1-t^2}\ dt.$ (2) Find the value of  $M$  such that  $f(x)$  has maximum value  $M$  at  $x=\frac{\sqrt{3}}{2a}.$ 

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