
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 vectors in the plane $\overset{→}{OA}=(-1,-3)$, $\overset{→}{OB}=(5,3)$, and $\overset{→}{OM}=(2,2)$, point $P$ is on line $OM$, and $\overset{→}{PA} \cdot \overset{→}{PB}=-16$.  
$(1)$ Find the coordinates of $\overset{→}{OP}$.  
$(2)$ Find the cosine of $\angle APB$.  
$(3)$ Let $t \in \mathbb{R}$, find the minimum value of $|\overset{→}{OA}+t\overset{→}{OP}|$.

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