For any two points \( M \) and \( N \) in the \( XY \)-plane, let \( \overrightarrow{MN} \) denote the vector from \( M \) to \( N \), and \( \vec{0} \) denote the zero vector. Let \( P. Q \) and \( R \) be three distinct points in the \( XY \)-plane. Let \( S \) be a point inside the triangle \( \Delta PQR \) such that \( \overrightarrow{SP}+5\overrightarrow{SQ}+6\overrightarrow{SR}=\vec{0} \).
Let \( E \) and \( F \) be the mid-points of the sides \( PR \) and \( QR \), respectively. Then the value of \(\displaystyle \frac{\text{length of the line segment } EF}{\text{length of the segment } ES}\) is ________.
