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

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