
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 right triangle \( ABC \) with a right angle at vertex \( C \) and leg lengths in the ratio of \( 1:3 \). Points \( K \) and \( L \) are the centers of squares that share one side with leg \( AC \) and \( BC \) respectively, and these squares do not overlap with triangle \( ABC \). Point \( M \) is the midpoint of the hypotenuse \( AB \).

a) Justify that point \( C \) lies on the line segment \( KL \).

b) Calculate the ratio of the areas of triangles \( ABC \) and \( KLM \).

(J. Švrček)

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