
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 $f(x) = x^{3} + 3ax^{2} + bx + a^{2}$ has a local extremum of $0$ at $x = -1$.  
$(1)$ Find the analytical expression of the function $f(x)$.  
$(2)$ Let $h(x) = f(x) - m + 1$. If the function $h(x)$ has three zeros, find the range of real number $m$.

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