
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
Around a hook with a worm in the same plane, a crucian carp and a gudgeon are swimming along two circles. In the indicated plane, a rectangular coordinate system is introduced where the hook (the common center of the circles) is at the point \((0, 0)\). At the initial moment, the crucian carp and the gudgeon are at the points \(M_{0}(-1, 2\sqrt{2})\) and \(N_{0}(2, -4\sqrt{2})\) respectively. The speed of the crucian carp is two and a half times that of the gudgeon, and both are moving clockwise. Determine the coordinates of all positions of the gudgeon at which the distance between the two fish will be the shortest.

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