
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 triangle \( \triangle ABC \), \( 3BC = AB + CA \). Let \( T \) be a point on side \( AC \) such that \( 4AT = AC \). Points \( K \) and \( L \) are on sides \( AB \) and \( CA \) respectively, with \( KL \parallel BC \). The line \( KL \) is tangent to the incircle of \( \triangle ABC \). Find the ratio of the segments into which \( BT \) divides \( KL \).

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