
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 is a triangle  $ABC$  with the property that  $|AB| + |AC| = 3|BC|$ . Let  $T$  be the point on segment  $AC$  such that  $|AC| = 4|AT|$ . Let  $K$  and  $L$  be points on the interior of line segments  $AB$  and  $AC$  respectively such that  $KL \parallel BC$  and  $KL$  is tangent to the inscribed circle of  $\vartriangle ABC$ . Let  $S$  be the intersection of  $BT$  and  $KL$ . Determine the ratio  $\frac{|SL|}{|KL|}$ 

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