
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  $ABCD$  be a square with side length  $2$ .  Let  $M$  and  $N$  be the midpoints of  $\overline{BC}$  and  $\overline{CD}$  respectively, and let  $X$  and  $Y$  be the feet of the perpendiculars from  $A$  to  $\overline{MD}$  and  $\overline{NB}$ , also respectively.  The square of the length of segment  $\overline{XY}$  can be written in the form  $\tfrac pq$  where  $p$  and  $q$  are positive relatively prime integers.  What is  $100p+q$ ?

*Proposed by David Altizio*

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