
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 $h(x)=(x-a)e^{x}+a$.

$(1)$ If $x \in [-1,1]$, find the minimum value of the function $h(x)$;

$(2)$ When $a=3$, if for $\forall x_{1} \in [-1,1]$, $\exists x_{2} \in [1,2]$, such that $h(x_{1}) \geqslant x_{2}^{2}-2bx_{2}-ae+e+ \frac {15}{2}$ holds, find the range of $b$.

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