
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 $C: y^2 = 2px$ ($p > 0$) with focus $F$, and a point $P(1, m)$ on $C$ such that $|PF| = 2$.
(1) If the ellipse $C′: \frac {x^{2}}{4} + \frac {y^{2}}{n} = 1$ shares the same focus as the parabola $C$, find the equation of the ellipse $C'$;
(2) Suppose that the parabola $C$ and the ellipse $C'$ from part (1) intersect at points $A$ and $B$. Find the equation of the hyperbola with asymptotes along the lines $OA$ and $OB$ and passing through point $P$.

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