Let $T  =  TNFTPP$,  and  let  $S$  be  the  sum  of  the  digits  of  $T$.  Cyclic  quadrilateral  $ABCD$  has  side lengths  $AB  =  S  -  11$,  $BC  =  2$,  $CD  =  3$,  and  $DA  =  10$.  Let  $M$  and  $N$  be  the  midpoints  of  sides  $AD$  and $BC$.  The  diagonals  $AC$  and  $BD$  intersect  $MN$  at  $P$  and  $Q$  respectively.  $\frac{PQ}{MN}$  can  be  expressed  as $\frac{m}{n}$  where  $m$  and  $n$  are  relatively  prime  positive  integers.  Determine  $m  +  n$.

[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, [hide=here's the value of T.]$T=2378$[/hide]

1. **Determine the value of \( S \):**
   Given \( T = 2378 \), we need to find the sum of the digits of \( T \).
   \[
   S = 2 + 3 + 7 + 8 = 20
   \]

2. **Calculate the side length \( AB \):**
   Given \( AB = S - 11 \),
   \[
   AB = 20 - 11 = 9
   \]

3. **Verify the side lengths of the cyclic quadrilateral \( ABCD \):**
   \[
   AB = 9, \quad BC = 2, \quad CD = 3, \quad DA = 10
   \]

4. **Find the midpoints \( M \) and \( N \):**
   - \( M \) is the midpoint of \( AD \).
   - \( N \) is the midpoint of \( BC \).

5. **Use the properties of cyclic quadrilaterals and midlines:**
   In a cyclic quadrilateral, the line segment joining the midpoints of opposite sides is parallel to the line segment joining the other pair of opposite sides and is half its length.

6. **Calculate the length of \( MN \):**
   Since \( M \) and \( N \) are midpoints,
   \[
   MN = \frac{1}{2} \times AC
   \]

7. **Calculate the length of \( AC \):**
   Using the Ptolemy's theorem for cyclic quadrilateral \( ABCD \):
   \[
   AC \cdot BD = AB \cdot CD + AD \cdot BC
   \]
   Let \( BD = x \),
   \[
   AC \cdot x = 9 \cdot 3 + 10 \cdot 2 = 27 + 20 = 47
   \]
   Since \( AC = \sqrt{94} \),
   \[
   \sqrt{94} \cdot x = 47 \implies x = \frac{47}{\sqrt{94}} = \frac{47 \sqrt{94}}{94} = \frac{47}{\sqrt{94}} \cdot \frac{\sqrt{94}}{\sqrt{94}} = \frac{47 \sqrt{94}}{94}
   \]

8. **Calculate \( PQ \):**
   Since \( PQ \) is the segment of \( MN \) intersected by diagonals \( AC \) and \( BD \), and using the properties of midlines in cyclic quadrilaterals,
   \[
   PQ = \frac{1}{2} \times BD
   \]
   \[
   PQ = \frac{1}{2} \times \frac{47 \sqrt{94}}{94} = \frac{47 \sqrt{94}}{188}
   \]

9. **Calculate the ratio \( \frac{PQ}{MN} \):**
   \[
   \frac{PQ}{MN} = \frac{\frac{47 \sqrt{94}}{188}}{\frac{\sqrt{94}}{2}} = \frac{47 \sqrt{94}}{188} \times \frac{2}{\sqrt{94}} = \frac{47 \times 2}{188} = \frac{94}{188} = \frac{1}{2}
   \]

10. **Express the ratio in simplest form:**
    \[
    \frac{PQ}{MN} = \frac{1}{2}
    \]
    Here, \( m = 1 \) and \( n = 2 \).

11. **Sum \( m \) and \( n \):**
    \[
    m + n = 1 + 2 = 3
    \]

The final answer is \(\boxed{3}\).