
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 the parabola $C$: $y^{2}=4x$ with focus $F$ and a point $P(1,m)$ on parabola $C$.
(1) If the ellipse $C'$: $\frac{x^{2}}{4}+ \frac{y^{2}}{n}=1$ shares a common focus with the parabola $C$, find the equation of ellipse $C'$.
(2) Let $A$ and $B$ be the intersection points of the parabola $C$ and the ellipse $C'$ found in part (1). Find the equation of the hyperbola that passes through point $P$, and has the lines containing segments $OA$ and $OB$ as its asymptotes.

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