
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
For the function $f(x)$, if there exists a real number $x$ in its domain such that $f(-x) = -f(x)$, then $f(x)$ is called a "local odd function".  
$(1)$ If $f(x) = 2^{x} + m$ is a "local odd function" defined on the interval $[-2,2]$, find the range of real number $m$;  
$(2)$ If $f(x) = 4^{x} + m2^{x+1} + m^{2} - 4$ is a "local odd function" defined on the domain $R$, find the range of real number $m$.

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