
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
Two identical equilateral triangles $ABC$ and $CDE$ with side length 1 are positioned such that they share exactly one common point $C$ and the angle $BCD$ is less than $60^{\circ}$. Point $K$ is the midpoint of $AC$, point $L$ is the midpoint of $CE$, and point $M$ is the midpoint of $BD$. The area of triangle $KLM$ is $\frac{\sqrt{3}}{5}$. Find $BD$.

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