
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 that $f(x)$ is an odd function, $g(x)$ is an even function, and $f(x) + g(x) = 2\log_2(1 - x)$.

1. Find the analytical expressions and domains of $f(x)$ and $g(x)$.
2. If the function $F(x) = 2^{g(x)} + (k + 2)x$ is monotonic in the interval $(-1, 1)$, find the range of the real number $k$.
3. If the equation $f(2^x) - m = 0$ has a solution, find the range of the real number $m$.

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