
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 $f(x) = x^3 - 6x^2 + ax + b$ ($a \in \mathbb{R}$), the equation of the tangent line at its intersection points with the $x$-axis is $y = 9x - 18$.
$(1)$ Find the expression for $f(x)$;
$(2)$ If $\frac{1}{10}kx(x - 2)^2 < f(x) < 9x + k$ holds for all $x \in (2,5)$, find the range of values for $k$.

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