
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 $f(x)=\sin (2x+\varphi)$, where $\varphi$ is a real number. If $f(x)\leqslant \left| f\left(\frac{\pi }{6}\right) \right|$ holds for all $x\in\mathbb{R}$, and $f\left(\frac{\pi }{2}\right) > f(\pi )$, find the interval of monotonic increase for $f(x)$.

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