
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 $f(x)= \frac{a\cdot 2^{x}+a-2}{2^{x}+1}$ ($x\in\mathbb{R}$), if $f(x)$ satisfies $f(-x)+f(x)=0$,
(1) find the value of the real number $a$ and $f(3)$;
(2) determine the monotonicity of the function and provide a proof.

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