
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
The vertices of a triangle have coordinates \(A(1 ; 3.5)\), \(B(13.5 ; 3.5)\), and \(C(11 ; 16)\). We consider horizontal lines defined by the equations \(y=n\), where \(n\) is an integer. Find the sum of the lengths of the segments cut by these lines on the sides of the triangle.

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