
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

Three swimmers must swim in a pool on a lane 50 meters long, immediately turn back, and return to the starting point. The first swimmer starts first, the second swimmer starts after $a$ seconds, and the third swimmer starts after another $a$ seconds. At a certain moment, while not having reached the end of the lane, the swimmers were equidistant from the starting point. The third swimmer, after reaching the end of the lane and turning back, met the second swimmer at $s$ meters from the end of the lane, and the first swimmer at $r$ meters from the end of the lane. Determine the speeds of the first and third swimmers and establish the conditions on $s$ and $r$ for which the problem has a solution.

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