
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
(12 points) The vertex of the parabola is at the origin, and its directrix passes through one of the foci of the hyperbola $x^2 - y^2 = 1$ ($a > 0$, $b > 0$), and is perpendicular to the real axis of the hyperbola. Given that the intersection point of the parabola and the hyperbola is $M$, find the equations of the parabola and the hyperbola.

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