
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
Among the following four propositions about the function $f(x)$: 

(1) If the function $f(x)$ is increasing, then the equation $f(x) = 0$ definitely has a solution.
(2) If the function $f(x)$ is decreasing, then the equation $f(x) = 0$ has at most one solution.
(3) If the function $f(x)$ is even, then the equation $f(x) = 0$ definitely has an even number of solutions.
(4) If the function $f(x)$ is odd, and the equation $f(x) = 1$ has a solution, then the equation $f(x) = -1$ also has a solution.

The correct propositions are:

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