
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 $a > 0$ and $a \neq 1$, let the function $f(x) = \log_{a}(x+1)$, and $g(x) = \log_{a} \frac{1}{1-x}$. Denote $F(x) = 2f(x) + g(x)$.

(1) Find the domain $D$ of function $F(x)$ and its zeros;
(2) Discuss the monotonicity of function $F(x)$ on its domain $D$;
(3) If the equation $F(x) - 2m^2 + 3m + 5 = 0$ has only one solution in the interval $[0, 1)$, find the range of the real number $m$.

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