
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 shares common foci $F_1$ and $F_2$ with the ellipse $\dfrac {x^{2}}{9}+ \dfrac {y^{2}}{25}=1$, and their sum of eccentricities is $2 \dfrac {4}{5}$.  
$(1)$ Find the standard equation of the hyperbola;  
$(2)$ Let $P$ be a point of intersection between the hyperbola and the ellipse, calculate $\cos \angle F_{1}PF_{2}$.

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