
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 that $\overrightarrow{OP} =(2,1)$, $\overrightarrow{OA} =(1,7)$, $\overrightarrow{OB} =(5,1)$, and let $M$ be a point on the line $OP$, with $O$ being the origin of the coordinate system.

(1) Find the $\overrightarrow{OM}$ that minimizes $\overrightarrow{MA} \cdot \overrightarrow{MB}$;

(2) For the point $M$ found in (1), find the cosine value of $\angle AMB$.

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