
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 vertex of a parabola is at the origin, its directrix passes through a focus of the hyperbola $\frac{x^{2}}{a^{2}} - \frac{y^{2}}{b^{2}} = 1$ $(a > 0, b > 0)$. Moreover, this directrix is perpendicular to the line connecting the two foci of the hyperbola. The parabola and the hyperbola intersect at point $P\left(\frac{3}{2}, \sqrt{6}\right)$. Find the equations of the parabola and the hyperbola.

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