Too much radius, not enough circle

A particle of mass 'm' starts performing circular motion on a vertical plane when given a an horizontal velocity U at the lowest point..

This Particle is held by a string of radius r which is fixed from one end. The fixed end is at a height 2r from the ground.

After completing the circle 7 times, the particle gets released at an angle $\theta$ above the horizontal in which the fixed end is present, The particle then performs a projectile motion where the Max height from the ground is 51r/13 , and the max height from the point of projection is r .

The particle hits the ground with a * velocity 50m/s, making an angle of 60 degree with the vertical *

Find the minimum length of the string required for this phenomenon to occur. Also find U. And find $sin \theta$

Give your answer in the form $sin\theta$ +|U| + r

Assumptions: Take g=10m/s^2 Neglect air resistance. Round off your 'r' to no decimal places.