
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
In a right triangle \( KLM \), a segment \( MD \) is drawn from the right-angle vertex to point \( D \) on the hypotenuse \( KL \) such that the lengths of the segments \( DL \), \( DM \), and \( DK \) are different and form a geometric progression in the given order with a common ratio of \( \sqrt{2} \), with \( DL = 1 \). Find the measure of the angle \( KMD \).

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