
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   $T  =  TNFTPP$ ,  and  let   $S$   be  the  sum  of  the  digits  of   $T$ .  In  triangle   $ABC$ ,  points   $D$ ,   $E$ ,  and   $F$   are  the feet  of  the  angle  bisectors  of   $\angle A$ ,   $\angle B$ ,   $\angle C$   respectively.  Let  point   $P$   be  the  intersection  of segments   $AD$   and   $BE$ ,  and  let   $p$   denote  the  perimeter  of   $ABC$ .  If   $AP  =  3PD$ ,   $BE  =  S  -  1$ ,  and   $CF  =  9$ , then  the  value  of   $\frac{AD}{p}$   can  be  expressed  uniquely  as  $\frac{\sqrt{m}}{n}$   where   $m$   and   $n$   are  positive  integers such  that   $m$   is  not  divisible  by  the  square  of  any  prime.  Find   $m  +  n$ .

[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>$T=1801$</details>.

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