
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 an ellipse $C\_1$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$ and a hyperbola $C\_2$ with common left and right foci $F\_1$ and $F\_2$. The product of the eccentricities of the two curves is $e\_1 \cdot e\_2 = 1$. $D$ is the intersection point of the two curves in the first quadrant, and $F\_1D$ intersects the $y$-axis at point $E$. Find the length of $EF\_2$ in terms of $a$ and $b$.

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