
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 $f(x)=(x+a)\ln(x+a)$, $g(x)=-\frac{a}{2}x^2+ax$.
$(1)$ For the function $h(x)=f(e^x-a)+g'(e^x)$, where $x \in [-1,1]$, find the minimum value of the function $h(x)$;
$(2)$ For any $x \in [2,+\infty)$, if $f(x-a-1)-g(x) \leqslant 0$ holds, find the range of $a$.

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