
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 the hyperbola $C$ has the same foci as the ellipse $\frac{x^{2}}{35}+ \frac{y^{2}}{10}=1$, and its asymptotes are $y=± \frac{4}{3}x$.
(1) Find the standard equation of the hyperbola $C$;
(2) Let $F_{1}$ be the left focus of the hyperbola, $P$ be a point on the right branch of the hyperbola $C$, and the midpoint of segment $PF_{1}$ lies on the $y$-axis, find the area of $\triangle PF_{1}F_{2}$.

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