
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 $F_{1}$ and $F_{2}$ are the left and right foci of the hyperbola $C$: $\frac{{x}^{2}}{{a}^{2}}-\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt 0,b \gt 0\right)$, and the hyperbola $C$ passes through the point $G(\frac{\sqrt{15}}{3}$,$\frac{2\sqrt{3}}{3}$), with $\overrightarrow{GF_{1}}\cdot \overrightarrow{GF_{2}}=0$. <br/>$(1)$ Find the equation of the hyperbola $C$; <br/>$(2)$ Given a line $l$ passing through the point $\left(0,-1\right)$ intersects the left and right branches of the hyperbola $C$ at points $M$ and $N$, and intersects the asymptotes of the hyperbola $C$ at points $P$ and $Q$ (point $Q$ is located to the right of the $y$-axis), find the range of $\frac{MP}{PQ}+\frac{QN}{PQ}$.

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