
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
Let   $T  =  TNFTPP$ .  Triangle   $ABC$   has  integer  side  lengths,  including   $BC  =  100T  -  4$ ,  and  a  right angle,   $\angle ABC$ .  Let   $r$   and   $s$   denote  the  inradius  and  semiperimeter  of   $ABC$   respectively.  Find  the ''perimeter''  of  the  triangle  ABC  which  minimizes   $\frac{s}{r}$ .

[b]Note: This is part of the Ultimate Problem, where each question depended on the previous question.  For those who wanted to try the problem separately, <details><summary>here's the value of T.</summary>$T=7$</details>

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