
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, let $P(x,y)$ be a point on the plane, and let $M(1,0)$. The distance from point $P$ to the line $x=2$ is $d$, and $\frac{|PM|}{d}=\frac{\sqrt{2}}{2}$.

(Ⅰ) Find the equation of the trajectory $C$ of point $P$;

(Ⅱ) A line $l$ that does not pass through the origin $O$ intersects $C$ at points $A$ and $B$, the midpoint of segment $AB$ is $D$, and the ordinate of the intersection of line $OD$ with the line $x=2$ is $1$. Find the maximum area of $\Delta OAB$ and the equation of line $l$ at this time.

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