
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 points O(0, 0), M(1, 0), and a circle C: $(x-5)^2+(y-4)^2=r^2$ ($r>0$) with at least one point P on it such that $|PO| = \sqrt{2}|PM|$, find the minimum value of $r$.

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