
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 a moving point $P$ on the circle $x^{2}+4x+y^{2}-32=0$, and the midpoint of the line segment connecting it to the fixed point $Q(6,0)$ is $M$.  
$(1)$ Find the equation of the locus of point $M$;  
$(2)$ The line $l$ passing through the fixed point $(0,-3)$ intersects the locus of point $M$ at two distinct points $A(x_{1},y_{1})$ and $B(x_{2},y_{2})$, satisfying $\frac{x_{1}}{x_{2}}+\frac{x_{2}}{x_{1}}=\frac{21}{2}$. Find the equation of the line $l$.

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