
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 $E$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with eccentricity $\frac{\sqrt{5}}{3}$, where the upper and lower vertices are $A$ and $B$ respectively, and the right vertex is $C$, and the area of $\triangle ABC$ is $6$. <br/>$(1)$ Find the equation of $E$;<br/>$(2)$ If point $P$ is a point on $E$ other than the vertices, and the line passing through $AP$ and $BC$ intersects at point $M$, and the line passing through $CP$ intersects the $y$-axis at point $N$, determine whether the line $MN$ passes through a fixed point. If so, find the coordinates of that fixed point; if not, explain the reason.

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