
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 a Cartesian coordinate system, the curve $C$ consists of all points for which the product of distances to the fixed points $A(1,0)$ and $B(-1,0)$ is equal to $\sqrt{2}$. Regarding curve $C$, the following four statements are given:

1. The curve $C$ is symmetric with respect to the axes;    
2. The curve $C$ is symmetric with respect to the origin;
3. All points on curve $C$ are located inside the unit circle $x^2 + y^2 = 1$;
4. The ordinates (y-coordinates) of all points on curve $C$ are within the interval $\left[-\frac{1}{2}, \frac{1}{2}\right]$.

Among these statements, the correct ones are ______.

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