
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
Given two points, \( C \) and \( D \), on the semicircle with diameter \( AB \), let point \( M \) move along the arc \( CD \) of the semicircle. Let \( S \) and \( T \) be the intersection points of segments \( MA \) and \( MB \) with segment \( CD \), respectively. Construct the position of \( M \) when the segment \( ST \) is the longest possible.

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