
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
Rachel does jumps, each of which is \(168 \mathrm{~cm}\) long. Joel does jumps, each of which is \(120 \mathrm{~cm}\) long. Mark does jumps, each of which is \(72 \mathrm{~cm}\) long.

(a) On Monday, Rachel completes 5 jumps and Joel completes \(n\) jumps. Rachel and Joel jump the same total distance. Determine the value of \(n\).

(b) On Tuesday, Joel completes \(r\) jumps and Mark completes \(t\) jumps. Joel and Mark jump the same total distance, and \(r\) and \(t\) are integers. If \(11 \leq t \leq 19\), determine the values of \(r\) and \(t\).

(c) On Wednesday, Rachel completes \(a\) jumps, Joel completes \(b\) jumps, and Mark completes \(c\) jumps. Each of \(a, b\) and \(c\) is a positive integer, and Rachel, Joel and Mark each jump the same total distance. Determine the minimum possible value of \(c\) and explain why this value is indeed the minimum.

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