
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 the function $f(x) = \sin x + \cos x + \sin x \cos x$, and $g(x) = m \cos(2x - \frac{\pi}{6}) - 2m + 3$ (with $m > 0$). If for any $x_1 \in [0, \frac{\pi}{4}]$, there exists $x_2 \in [0, \frac{\pi}{4}]$ such that $g(x_1) = f(x_2)$ holds, determine the range of the real number $m$.

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