
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

Two swimmers are training in a rectangular quarry. The first swimmer prefers to start at the corner of the quarry, so he swims diagonally to the opposite corner and back. The second swimmer prefers to start at a point that divides one of the sides of the quarry in the ratio \(2018: 2019\). He swims along a quadrilateral, visiting one point on each side, and returns to the starting point. Can the second swimmer choose points on the other three sides in such a way that his path is shorter than the first swimmer's? What is the minimum value that the ratio of the length of the longer path to the shorter one can have?

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