
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
If a function $f(x)$ has real numbers $x$ in its domain such that $f(-x)=-f(x)$, then $f(x)$ is called a "locally odd function".
$(1)$ When the domain is $[-1,1]$, determine whether $f(x)=x^{4}+x^{3}+x^{2}+x-1$ is a "locally odd function";
$(2)$ If $g(x)=4^{x}-m\cdot2^{x+1}+m^{2}-3$ is a "locally odd function" over the domain $\mathbb{R}$, find the range of the real number $m$;
$(3)$ Given $a > 1$, for any $b\in\left[1, \frac{3}{2}\right]$, the function $h(x)=\ln(x+1+a)+x^{2}+x-b$ is a "locally odd function" over the domain $[-1,1]$, find the range of the real number $a$.

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