
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with a minor axis length of $2$, point $P$ lies on ellipse $C$, and the maximum area of the triangle formed by $P$ and the two foci is $\sqrt{3}$. <br/>$(Ⅰ)$ Find the standard equation of ellipse $C$; <br/>$(Ⅱ)$ When $P$ is the right vertex of ellipse $C$, the line $l$ intersects ellipse $C$ at points $M$ and $N$ (other than point $P$), and $PM\perp PN$. Determine whether the line $l$ passes through a fixed point. If it does, 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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