
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 $f(x) = \sqrt{3}\sin x\cos x + \cos^2 x + a$.

(1) Find the smallest positive period and the monotonically increasing interval of the function $f(x)$;

(2) When $x \in [-\frac{\pi}{6}, \frac{\pi}{3}]$, the sum of the maximum and minimum values of the function $f(x)$ is $\frac{3}{2}$. Find the value of the real number $a$.

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