
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 $f(x)=ax^3+bx^2-3x$ $(a,b \in \mathbb{R})$, where $f'(x)$ is the derivative of $f(x)$, if $f'(x)$ is an even function and $f'(1)=0$,
1. Find the analytical expression of $f(x)$;
2. If for any two arguments $x_1$, $x_2$ in the interval $[1, 2]$, it holds that $|g(x_1) - g(x_2)| \leq c$, where $g(x) = \frac{1}{3}f(x) - 6\ln x$, find the minimum value of the real number $c$;
3. If there are three tangent lines that can be drawn through point M(2, m) to the curve $y=xf(x)$, find the range of values for the real number $m$.

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