
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 $f(x)=x\ln x$, $g(x)=ax^2 - x$ (where $a \in \mathbb{R}$),
1. Determine the monotonic intervals and extremum points of $f(x)$.
2. Find the range of real numbers $a$ such that $f(x) \le g(x)$ holds true for all $x$.
3. When $a = \frac{1}{8}$, determine if there exists a real number $m$ such that the equation $\frac{3f(x)}{4x} + m + g(x) = 0$ has three distinct real roots. If such an $m$ exists, find the range of $m$; if not, explain why.

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