
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
A ellipse with its center at the origin and foci on the x-axis shares the same foci $F\_1$, $F\_2$ with a hyperbola, and $|F\_1F\_2|=2\sqrt{13}$. The difference between the semi-major axis of the ellipse and the real semi-axis of the hyperbola is 4, and the ratio of their eccentricities is $\frac{3}{7}$.

1. Find the equations of these two curves.
2. If $P$ is one of the intersection points of these two curves, find the value of $\cos∠F\_1PF\_2$.

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