
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 $C$ be an arbitrary point on the circle $k$ with diameter $AB$, different from points $A$ and $B$. Draw the perpendicular from point $C$ to the diameter $AB$, with the foot of the perpendicular being point $D$ on segment $AB$, and let the second intersection of this perpendicular with circle $k$ be point $E$. The circle with center $C$ and radius $CD$ intersects the circle $k$ at points $P$ and $Q$. Let $M$ be the intersection point of segments $CE$ and $PQ$. Determine the value of $\frac{P M}{P E}+\frac{Q M}{Q E}$.

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