
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 find the domain $D$ of 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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