
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-angled triangle, the altitude and median are drawn from the vertex of the right angle. The ratio of their lengths is $40:41$. Find the ratio of the lengths of the legs.

Given: $\triangle ABC, \widehat{ACB}=90^{\circ}, [CK]$ is the altitude, $[CM]$ is the median, $|CK|:|CM| = 40:41$.

Find: $|AC|:|BC|$.

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