
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
Suppose the solution set of the equation $f(x, y) = 0$ is non-empty. If the statement "All points with coordinates satisfying the equation $f(x, y) = 0$ are on the curve $C$" is incorrect, consider the following five statements:
1. All points with coordinates satisfying $f(x, y) = 0$ are not on the curve $C$.
2. The coordinates of all points on the curve $C$ do not satisfy $f(x, y) = 0$.
3. Not all points with coordinates satisfying $f(x, y) = 0$ are on the curve $C$.
4. There must be points not on the curve $C$, whose coordinates satisfy $f(x, y) = 0$.
5. Some points with coordinates satisfying $f(x, y) = 0$ are on the curve $C$, and some are not.

Which of the above statements are correct? (Fill in the numbers of all correct statements.)

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