
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
In trapezoid  $ABCD$ ,  $AB < CD$ ,  $AB\perp BC$ ,  $AB\parallel CD$ , and the diagonals  $AC$ ,  $BD$  are perpendicular at point  $P$ . There is a point  $Q$  on ray  $CA$  past  $A$  such that  $QD\perp DC$ . If
\[\frac{QP} {AP}+\frac{AP} {QP} = \left( \frac{51}{14}\right)^4 - 2,\]then  $\frac{BP} {AP}-\frac{AP}{BP}$  can be expressed in the form  $\frac{m}{n}$  for relatively prime positive integers  $m,n$ . Compute  $m+n$ .

*Ray Li.*

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