
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 acute triangle \(ABC\), let \(D, E\), and \(F\) be the feet of the altitudes from \(A, B\), and \(C\) respectively, and let \(L, M\), and \(N\) be the midpoints of \(BC, CA\), and \(AB\), respectively. Lines \(DE\) and \(NL\) intersect at \(X\), lines \(DF\) and \(LM\) intersect at \(Y\), and lines \(XY\) and \(BC\) intersect at \(Z\). Find \(\frac{ZB}{ZC}\) in terms of \(AB, AC\), and \(BC\).

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