
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
In the Cartesian coordinate system, point $P$ is on the ellipse $C$ with an eccentricity of $\frac{1}{2}$. $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$, and the perimeter of $\triangle F_{1}PF_{2}$ is $6$.

(Ⅰ) Find the standard equation of the ellipse $C$;

(Ⅱ) Given two fixed points in the plane, $A(-2,0)$ and $B(2,0)$, and a line $l$ with equation $x=my+1$ ($m\in R$) intersects the ellipse $C$ at points $M$ and $N$. Determine whether the intersection point of lines $AM$ and $BN$ is on a straight line. If so, find the equation of this line; otherwise, explain why.

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