
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  $T  =  TNFTPP$ .  When  properly  sorted,   $T  -  35$   math  books  on  a  shelf  are  arranged  in  alphabetical order  from  left  to  right.  An  eager  student  checked  out  and  read  all  of  them.  Unfortunately,  the student  did  not  realize  how  the  books  were  sorted,  and  so  after  finishing  the  student  put  the  books  back  on  the  shelf  in  a  random  order.  If  all  arrangements  are  equally  likely,  the  probability  that exactly   $6$   of  the  books  were  returned  to  their  correct  (original)  position  can  be  expressed  as   $\frac{m}{n}$ ,  where   $m$   and   $n$   are  relatively  prime  positive  integers.  Compute   $m  +  n$ .

[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>$T=44$</details>.

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