
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
Let $x \in [0, \pi]$. Find the intervals where the function of $x$ is monotonically increasing.  
After shifting the graph of $f(x)$ to the left by $\frac{\pi}{6}$ units, we obtain the graph of $g(x)$. If the function $g(x) - m$ has a zero in the interval $\left[0, \frac{\pi}{2}\right]$, find the range of the real number $m$.

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