
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
A carp and a gudgeon are swimming around a hook with a worm along two circles. The circles are on the same plane, with the hook (the common center of the circles) located at the point $(0, 0)$ in a rectangular coordinate system. Initially, the 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 carp is two and a half times the speed 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 is greatest.

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