
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 function $f(x)=\sin 2x+a\cos x+x$ reaches an extreme value at the point $x=\frac{\pi }{6}$.
(1) Find the value of the real number $a$;
(2) Find the maximum value of the function $f(x)$ when $x\in\left[-\frac{\pi }{6},\frac{7\pi }{6}\right]$.

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