
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
Along the water, around a float, a water strider and a diving beetle move counterclockwise along two circles. A rectangular coordinate system is introduced on the water surface, with the float (the common center of the circles) located at the point (0, 0). The speed of the water strider is twice the speed of the beetle. At the initial moment in time, the water strider and the beetle are at the points \(M_{0}(-2 ;-2 \sqrt{7})\) and \(N_{0}(5 ; 5 \sqrt{7})\) respectively. Determine the coordinates of all positions of the beetle where the distance between the two insects will be shortest.

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