
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
Arpon chooses a positive real number \( k \). For each positive integer \( n \), he places a marker at the point \((n, n k)\) in the \((x, y)\) plane. Suppose that two markers whose \( x \) coordinates differ by 4 have a distance of 31. What is the distance between the markers at \((7, 7 k)\) and \((19, 19 k)\)?

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