
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 that a hyperbola and an ellipse $\frac{x^{2}}{25}+\frac{y^{2}}{9}=1$ have a common focus, and the sum of their eccentricities is $\frac{14}{5}$.  
$(1)$ Find the equation of the hyperbola.  
$(2)$ A line $l$ passing through the point $P(3,1)$ intersects the hyperbola at the line segment $AB$ which is bisected by the point $P$. Find the equation of the line $l$.

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