Integer Lengths

Geometry Level 4

Given triangle \(ABC\), let \(P\) be a point on \(AB\) and \(Q\) be a point on \(AC\).

\(PQ \cap BC=X\) and \(\angle PXB=\angle C - \angle B\)

\(PC \times QB=1000\) and \(PB \times QC=371\).

Given that \(BC>PQ>1\) and \(BC\) and \(PQ\) are integers, find the length of \(BC\).

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