
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 $f(x) = a - \frac{2}{2^x + 1} (a \in \mathbb{R})$.
(1) Explore the monotonicity of the function $f(x)$;
(2) Does there exist a real number $a$ such that $f(x)$ is an odd function?

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