
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 functions $f(x)=x\ln x$ and $g(x)=ax^{2}-x$ where $a\in \mathbb{R}$. Find:

1. The intervals of monotonicity and critical points of $f(x)$.
2. The range of real numbers for $a$ that satisfies $f(x)\leq g(x)$.
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 so, find the range of values for $m$; if not, explain the reason.

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