
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
Consider the function $f(x) = \sin x \sin\left(\frac{\pi}{2} + x\right) + \cos^2 x$. In triangle $\triangle ABC$, the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively.
(1) Find the maximum value of $f(x)$.
(2) Given that $f(A) = 1$, $A+B = \frac{7\pi}{12}$, and $b = \sqrt{6}$, find the values of angle $A$ and side $a$.

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