
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
A parabola has its vertex at the origin, and its axis passes through one of the foci of the hyperbola $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1$. The parabola is perpendicular to the real axis of the hyperbola, and it intersects with the hyperbola at the point $(\frac{3}{2},\sqrt{6})$. Find the equations of the parabola and the hyperbola.

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