
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 a square $ABCD$ with a side length of $1$, a line $MN$ passes through the center $O$ of the square and intersects with the edges $AD$ and $BC$ at points $M$ and $N$ respectively. If point $P$ satisfies the condition $2\vec{OP} = l\vec{OA} + (1-l)\vec{OB}$ ($l \in \mathbb{R}$), find the minimum value of $\vec{PM} \cdot \vec{PN}$.

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