
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]


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Given a general triangle \(ABC\) with points \(K, L, M, N, U\) on its sides:

- Point \(K\) is the midpoint of side \(AC\).
- Point \(U\) is the midpoint of side \(BC\).
- Points \(L\) and \(M\) lie on segments \(CK\) and \(CU\) respectively, such that \(LM \parallel KU\).
- Point \(N\) lies on segment \(AB\) such that \(|AN| : |AB| = 3 : 7\).
- The ratio of the areas of polygons \(UMLK\) and \(MLKNU\) is 3 : 7.

Determine the length ratio of segments \(LM\) and \(KU\).

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