
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 it is called a "locally odd function".

(Ⅰ) Given the quadratic function $f(x)=ax^{2}+2x-4a$, determine whether $f(x)$ is a "locally odd function" and explain the reason;

(Ⅱ) If $f(x)=2^{x}+m$ is a "locally odd function" defined on the interval $[-1,1]$, find the range of the real number $m$;

(Ⅲ) If $f(x)=4^{x}-m\cdot2^{x+1}-3$ is a "locally odd function" whose domain is $\mathbb{R}$, find the range of the real number $m$;

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