
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
A quadrilateral \(KLMN\) is inscribed in a circle, with diagonals \(KM\) and \(LN\) intersecting at point \(T\). The feet of the perpendiculars dropped from point \(T\) to the sides of the quadrilateral lie on these sides. The distances from point \(T\) to the sides \(KL\), \(LM\), \(MN\), and \(NK\) are \(4 \sqrt{2}\), \(\sqrt{2}\), \(\frac{8}{\sqrt{17}}\), and \(\frac{8}{\sqrt{17}}\) respectively.
a) Find the ratio \(KT: TM\).

b) Find the length of the diagonal \(LN\), given that \(KM = 10\).

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