
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 $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its left and right foci are $F_1$ and $F_2$ respectively, and its top vertex is $A$. Point $P(1, \frac{3}{2})$ is on the ellipse $C$. A line passing through point $A$ and perpendicular to $AF_2$ intersects the negative half of the x-axis at point $B$, and $2\vec{F_1F_2} + \vec{F_2B} = \vec{0}$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Is there a line $m$ passing through point $Q(4,0)$ that intersects the ellipse $C$ at two distinct points $M$ and $N$, such that $36|QP|^2 = 35|QM| \cdot |QN|$? If it exists, find the equation of line $m$; if not, explain why.

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