
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
In \(\triangle QRS\), point \(T\) is on \(QS\) with \(\angle QRT = \angle SRT\). Suppose that \(QT = m\) and \(TS = n\) for some integers \(m\) and \(n\) with \(n > m\) and for which \(n + m\) is a multiple of \(n - m\). Suppose also that the perimeter of \(\triangle QRS\) is \(p\) and that the number of possible integer values for \(p\) is \(m^2 + 2m - 1\). Determine the value of \(n - m\).

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