
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 sides of length $10$ inches, segment $AE$ is drawn where $E$ is the point on $DC$, $3$ inches from $D$. The perpendicular bisector of $AE$ is drawn and intersects $AE, AD$, and $BC$ at points $M, P$, and $Q$ respectively. Calculate the ratio of segment $PM$ to $MQ$.
A) $1:1$
B) $2:1$
C) $3:2$
D) $4:1$

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