
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 a rectangle \( \mathrm{ABCD} \). The side \( \mathrm{BC} \) is one and a half times the length of side \( \mathrm{AB} \). Point \( \mathrm{K} \) is the midpoint of side \( \mathrm{AD} \). Point \( \mathrm{L} \) on side \( \mathrm{CD} \) is such that \( \mathrm{CL} = \mathrm{AK} \). Point \( \mathrm{M} \) is the intersection of line \( \mathrm{BL} \) and the perpendicular bisector of segment \( \mathrm{DL} \). It is known that \( \mathrm{KL} = 4 \). Find the length of segment \( \mathrm{BM} \).

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