
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 $f(x) = \begin{cases} 1 & (1 \leq x \leq 2) \\ \frac{1}{2}x^2 - 1 & (2 < x \leq 3) \end{cases}$, for any real number $a$, let $h(a) = \max\{f(x) - ax \mid x \in [1, 3]\} - \min\{f(x) - ax \mid x \in [1, 3]\}$.

(1) Find $h(0) = \_\_\_\_\_\_$.

(2) Find the analytical expression of $h(a)$ and its minimum value.

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