
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
Given the ellipse $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1$ and the hyperbola $\dfrac{x^{2}}{a^{2}} - \dfrac{y^{2}}{b^{2}} = 1$ (where $a > b > 0$), the eccentricities of the ellipse and the hyperbola are $e_{1}$ and $e_{2}$, respectively. Consider the following statements:
1. $e_{1}e_{2} < 1$;
2. $e_{1}^{2} + e_{2}^{2} = 2$;
3. $e_{1}e_{2} > 1$;
4. $e_{1}e_{2} = 1$;
5. $e_{1} + e_{2} < 2$.

Which of the above statements are correct?

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