
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
In the Cartesian coordinate system, the curve $C$ consists of all points whose product of distances to two fixed points $A(1,0)$ and $B(-1,0)$ equals $2$. For curve $C$, there are four statements:
1. The curve $C$ is symmetric about the axes;
2. The curve $C$ is symmetric about the origin;
3. All points on curve $C$ are inside the unit circle $x^{2}+y^{2}=1$;
4. The y-coordinates of all points on curve $C$ are within $[-\frac{1}{2}, \frac{1}{2}]$.

Among these, the correct statements are \_\_\_\_\_\_.

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