
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
The domain of \( f(x) \) is \( \mathbf{R} \). If \( c \in \mathbf{R} \) such that \( f(c)=c \), \( c \) is called a fixed point of \( f(x) \). Suppose the number of fixed points of \( f(x) \) is finite. Determine whether the following statements are correct. If correct, provide a proof; if not, give a counterexample.

(1) If \( f(x) \) is an odd function, then the number of fixed points of \( f(x) \) is odd.
(2) If \( f(x) \) is an even function, then the number of fixed points of \( f(x) \) is even.
(Peking University enrollment exam for science experimental class, 1993).

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