
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 $f(x)=4^{x}-2^{x}$, and real numbers $s$, $t$ satisfy $f(s)+f(t)=0$, $a=2^{s}+2^{t}$, $b=2^{s+t}$.  
$(1)$ When the domain of function $f(x)$ is $[-1,1]$, find the range of $f(x)$;  
$(2)$ Find the function relation $b=g(a)$, and the domain $D$ of the function $g(a)$;  
$(3)$ Based on the conclusion in $(2)$, for any $x_{1} \in D$, there exists $x_{2} \in [-1,1]$ such that $g(x_{1})=f(x_{2})+m$ holds, find the range of the real number $m$.

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