
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 $f(x) = 4\ln x - mx^2 + 1$ where $m \in \mathbb{R}$.

(1) Discuss the monotonicity of the function $f(x)$.

(2) If for any $x \in [1, e]$, $f(x) \leq 0$ always holds, find the range of the real number $m$.

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