
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 in the Cartesian coordinate system, O is the origin, vector $\overrightarrow {OA} = (1,7)$, $\overrightarrow {OB} = (5,1)$, $\overrightarrow {OP} = (2,1)$, and point M is a moving point on line OP.  
(I) When $\overrightarrow {MA} \cdot \overrightarrow {MB}$ takes the minimum value, find the coordinates of vector $\overrightarrow {OM}$;  
(II) Under the condition that point M satisfies (I), find the cosine value of $\angle AMB$.

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