
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
Let ellipse $C:\frac{{x^2}}{{a^2}}+\frac{{y^2}}{{b^2}}=1(a>b>0),{F_1},{F_2}$ be the left and right foci, $B$ be the endpoint of the minor axis, the length of the major axis be $4$, the focal length be $2c$, and $b>c$, the area of $\triangle BF_{1}F_{2}$ is $\sqrt{3}$. <br/>$(Ⅰ)$ Find the equation of ellipse $C$. <br/>$(Ⅱ)$ Let the moving line be $l:y=kx+m$ and ellipse $C$ have exactly one common point $M$, and intersect with the line $x=4$ at point $N$. Investigate whether there exists a fixed point $P$ in the coordinate plane such that the circle with diameter $MN$ always passes through point $P$. If it exists, find the coordinates of point $P$, if not, explain the reason.

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