
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 an ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ with left and right vertices $M_{1}$, $M_{2}$, and a short axis length of $2\sqrt{3}$, a point $P$ on $C$ satisfies that the product of the slopes of lines $PM_{1}$ and $PM_{2}$ is $-\frac{3}{4}$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ If a line $l$ passing through point $\left(1,0\right)$ and not perpendicular to the $y$-axis intersects $C$ at points $A$ and $B$, and lines $M_{1}A$ and $M_{2}B$ intersect at point $Q$. Investigate whether point $Q$ lies on a fixed line. If yes, find the equation of the fixed line; if not, explain the reason.

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