An object is projected with a speed \(10\text{ ms}^{-1}\) at an angle of projection, \(\theta =\dfrac{\pi}{6}\) with the horizontal. The object breaks down into \(n\) fragments of equal mass during its motion. One fragment is found to strike the ground at a distance of \(\sqrt{3}\text{ m}\) from the point of projection in the same azimuthal plane, in which the object is projected. If the center of mass of the remaining fragments strike the ground at a distance of \(7\sqrt{3}\text{ m}\) from the point of projection, find the value of \(n\).

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