
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 the parabola $C$: $x^{2}=4y$ with focus $F$, a line passing through point $F$ with a slope of $1$ intersects the parabola at points $M$ and $N$. Let line $l$ be the tangent line of parabola $C$ and $l \parallel MN$. If $P$ is a point on line $l$, then the minimum value of $\overrightarrow{PM} \cdot \overrightarrow{PN}$ is \_\_\_\_\_\_.

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