
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 hyperbola $C$: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ ($a > 0, b > 0$) with asymptotic equations $y = \pm \sqrt{3}x$, and $O$ as the origin, the point $M(\sqrt{5}, \sqrt{3})$ lies on the hyperbola.

$(1)$ Find the equation of the hyperbola $C$.

$(2)$ If a line $l$ intersects the hyperbola at points $P$ and $Q$, and $\overrightarrow{OP} \cdot \overrightarrow{OQ} = 0$, find the minimum value of $|OP|^2 + |OQ|^2$.

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