
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 $a, b \in \mathbb{R}$, let $max\{a, b\} = \left\{\begin{array}{l}a, \text{ if } a \geq b \\b, \text{ if } a < b\end{array}\right.$, and the function $f(x) = \max \{|x+1|, |x-2|\}$ for $x \in \mathbb{R}$.  
$(1)$ Write down the analytical expression of $f(x)$ and find the minimum value of $f(x)$.  
$(2)$ If the function $g(x) = x^2 - kf(x)$ is monotonically increasing on $(-\infty, -1]$, find the range of values for $k$.

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