<p>The twelve letters <img alt="$A$" height="13" src="https://latex.artofproblemsolving.com/0/1/9/019e9892786e493964e145e7c5cf7b700314e53b.png" width="13" />,<img alt="$B$" height="12" src="https://latex.artofproblemsolving.com/f/f/5/ff5fb3d775862e2123b007eb4373ff6cc1a34d4e.png" width="14" />,<img alt="$C$" height="12" src="https://latex.artofproblemsolving.com/c/3/3/c3355896da590fc491a10150a50416687626d7cc.png" width="14" />,<img alt="$D$" height="14" src="https://latex.artofproblemsolving.com/9/f/f/9ffb448918db29f2a72f8f87f421b3b3cad18f95.png" width="17" />,<img alt="$E$" height="14" src="https://latex.artofproblemsolving.com/f/a/2/fa2fa899f0afb05d6837885523503a2d4df434f9.png" width="16" />,<img alt="$F$" height="14" src="https://latex.artofproblemsolving.com/a/0/5/a055f405829e64a3b70253ab67cb45ed6ed5bb29.png" width="16" />,<img alt="$G$" height="12" src="https://latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png" width="14" />,<img alt="$H$" height="12" src="https://latex.artofproblemsolving.com/b/1/9/b1902d279ba37d60bdce4e0e987b7cd19d48974e.png" width="16" />,<img alt="$I$" height="12" src="https://latex.artofproblemsolving.com/0/2/7/027f4a11d6090f9eac0ce2488df6384dad1263ea.png" width="9" />,<img alt="$J$" height="14" src="https://latex.artofproblemsolving.com/a/b/b/abb5588023cfa1ac14643e9778699f03eecc57a3.png" width="14" />,<img alt="$K$" height="12" src="https://latex.artofproblemsolving.com/d/f/b/dfb064112b6c94470339f6571f69d07afc1c024c.png" width="16" />, and <img alt="$L$" height="12" src="https://latex.artofproblemsolving.com/8/5/9/859ccf4cd60c7bc6b8fa1afc9a42dc811a826d6f.png" width="12" /> are randomly grouped into six pairs of letters. The two letters in each pair are placed next to each other in alphabetical order to form six two-letter words, and then those six words are listed alphabetically. For example, a possible result is <img alt="$AB$" height="13" src="https://latex.artofproblemsolving.com/5/7/e/57ee5125358c0606c9b588580ddfa66f83e607b7.png" width="27" />, <img alt="$CJ$" height="12" src="https://latex.artofproblemsolving.com/b/3/4/b34fde4f5d148a180241e972fb4671cbcb5b616a.png" width="26" />, <img alt="$DG$" height="12" src="https://latex.artofproblemsolving.com/e/a/f/eaf9409044fef7f3966027eec40b2cced52eb16c.png" width="29" />, <img alt="$EK$" height="12" src="https://latex.artofproblemsolving.com/5/5/a/55a81d02ca95dbec9524e0269d05fb86c86e2f49.png" width="31" />, <img alt="$FL$" height="12" src="https://latex.artofproblemsolving.com/8/c/c/8ccd6bdfd1c8b22e53cab5a58b25f0680f2fc62f.png" width="26" />, <img alt="$HI$" height="12" src="https://latex.artofproblemsolving.com/b/5/3/b53cb842327bbae670d7c27b43b91206af890094.png" width="26" />. The probability that the last word listed contains <img alt="$G$" height="12" src="https://latex.artofproblemsolving.com/6/e/2/6e28ce12d49d39f160d5a0ef54077fc98e4b9d2b.png" width="14" /> is <img alt="$\frac mn$" height="33" src="https://latex.artofproblemsolving.com/9/b/9/9b9ffa9646e74f6e94a4a66b824c3a3f08f4f4b9.png" width="18" />, where <img alt="$m$" height="8" src="https://latex.artofproblemsolving.com/f/5/0/f5047d1e0cbb50ec208923a22cd517c55100fa7b.png" width="15" /> and <img alt="$n$" height="8" src="https://latex.artofproblemsolving.com/1/7/4/174fadd07fd54c9afe288e96558c92e0c1da733a.png" width="10" /> are relatively prime positive integers. Find <img alt="$m+n$" height="12" src="https://latex.artofproblemsolving.com/0/2/d/02dbab3601fa6cce490d54263017048f69e1a35d.png" width="48" />.</p>