
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
Let $\overrightarrow {OA} = (5, 1)$, $\overrightarrow {OB} = (1, 7)$, $\overrightarrow {OC} = (4, 2)$, and $\overrightarrow {OM} = t \overrightarrow {OC}$.

(1) Does there exist a real number $t$ such that $\overrightarrow {MA} \perp \overrightarrow {MB}$? If it exists, find the real number $t$; if not, explain why.

(2) Find the coordinates of point $M$ that minimizes the value of $\overrightarrow {MA} \cdot \overrightarrow {MB}$.

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