
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
Inside square $ABCD$ with a side length of $10$ units, a segment $AE$ is drawn where $E$ is located $6$ units from $D$ on side $DC$. A perpendicular bisector of segment $AE$ is drawn, intersecting $AE$ at point $M$, and further intersecting $AD$ and $BC$ at points $P$ and $Q$ respectively. Calculate the ratio of segment $PM$ to segment $MQ$.
A) 5:12  
B) 5:13  
C) 1:1  
D) 1:4  
E) 5:21

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