
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  $(n,nk)$  in the  $(x,y)$  plane. Suppose that two markers whose  $x$ -coordinates differ by  $4$  have distance  $31$ . What is the distance between the markers  $(7,7k)$  and  $(19,19k)$ ?

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