
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 $f(x) = \sin (2x + \frac{\pi}{6}) + 2\sin^2x$,

$(1)$ Find the smallest positive period of the function $f(x)$;

$(2)$ Find the maximum value of the function $f(x)$ and the set of values of $x$ for which this maximum is achieved;

$(3)$ Find the intervals on which the function $f(x)$ is monotonically increasing.

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