
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 $M: y^2 = 12x$ with focus $F$, and the hyperbola $C: \frac{y^2}{a^2} - \frac{x^2}{b^2} = 1$ $(a>0, b>0)$ whose asymptotes are at a distance of $\frac{3\sqrt{10}}{4}$ from point $F$. Point $P$ is a moving point on parabola $M$, and the sum of the distance from $P$ to the focus $F_1(0,c)$ of hyperbola $C$ and the distance to the line $x=-3$ has a minimum value of 5. Derive the equation of hyperbola $C$.

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