
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
The function $f(x) = \sin(2x + \varphi)$, where $\varphi$ is a real number. If $f(x) \leq |f(\frac{\pi}{6})|$ holds true for $(0, +\infty)$, and $f(\frac{\pi}{2}) > f(\pi)$, then the interval of monotonic increase for $f(x)$ is \_\_\_\_\_\_.

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