
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, and a perpendicular line to line $l$ passes through point $P$ with foot point $Q$. 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) Find a point $M$ on the trajectory $C$ such that the distance from $M$ to the line $y = x - 3$ is the shortest, and find the shortest distance.

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