What's the point?

Geometry Level 4

The line \(\color{red}{l}\) has equations \[\dfrac{x-1}{2}=\dfrac{y-1}{3}=\dfrac{z+1}{2}\] and the point \(A\) is \((7, 3, 7)\). \(M\) is the point where the \(\color{#007fff}{\text{perpendicular}}\) from \(A\) meets \(\color{red}{l}\).

The point \(\color{#ff8600}{B}\) lies on \(AM\) such that \(\overrightarrow { AB } =3 \, \overrightarrow { BM } \). If the coordinates of \(\color{#ff8600}{B}\) are \((p, q, r)\), find the value of \(pqr\).

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