
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 points $F_{1}(-1,0)$, $F_{2}(1,0)$, and a moving point $M$ such that the distance from $M$ to $F_{2}$ is $2\sqrt{2}$. The perpendicular bisector of segment $MF_{1}$ intersects segment $MF_{2}$ at point $P$.

(Ⅰ) As point $M$ varies, find the equation of the trajectory $G$ of the moving point $P$;

(Ⅱ) A line $L$ passing through point $F_{2}$ and not coinciding with the $x$-axis intersects the curve $G$ at points $A$ and $B$. A line parallel to the $x$-axis passing through point $B$ intersects the line $x=2$ at point $C$. Determine whether the line $AC$ always passes through a fixed point, and if so, find this fixed point; if not, explain why.

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