
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 $xoy$, given point $F(0,1)$ and line $l: y=-1$, let $P$ be a moving point on the plane. A perpendicular line is drawn from point $P$ to line $l$, and the foot of the perpendicular is $Q$. It satisfies $\overrightarrow{QP} \cdot \overrightarrow{QF} = \overrightarrow{FP} \cdot \overrightarrow{FQ}$.

(Ⅰ) Find the equation of the trajectory $C$ of the moving point $P$;  
(Ⅱ) On the trajectory $C$, find a point $M$ such that the distance from $M$ to the line $y=x-3$ is the shortest, and find this shortest distance.

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