
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 $F(x)=e^{x}$ satisfies $F(x)=g(x)+h(x)$, and $g(x)$, $h(x)$ are respectively even and odd functions defined on $\mathbb{R}$.  
$(1)$ Find the inverse function of $h(x)$.  
$(2)$ Given $\varphi(x)=g(x-1)$, if the function $\varphi(x)$ satisfies $\varphi(2a+1) > \varphi(-\frac{a}{2})$ on the interval $[-1,3]$, find the range of the real number $a$.  
$(3)$ If for any $x \in (0,2]$ the inequality $g(2x)-ah(x) \geqslant 0$ always holds, find the range of the real number $a$.

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