
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 functions $$f(x)=\log_{2}\left(\frac{x-1}{x+1}\right)$$, $g(x)=3ax+1-a$, and $h(x)=f(x)+g(x)$.  
(1) When $a=1$, determine the monotonicity and the number of zeros of the function $h(x)$ on the interval $(1, +\infty)$.  
(2) If the equation $f(x)=\log_{2}g(x)$ has two distinct real roots, find the range of the real number $a$.

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