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Two very small mass particles $m_1$, $m_2$ are connected by a $2l$ long, infinitely soft and inelastic thread without mass.

The initial condition of the system before being freely released is as in the Figure:

The midpoint $O$ of the thread is exactly at the edge of a table. Suppose the acceleration of gravity is $g$, there is no friction between particle, thread and the table, and the partile $m_1$ freely drops in an absolute vacuum environment and no air resistance;

what is the locus of particle $m_1$ before it hits the table?

I obtained a complicate nonlinear system of ordinary differential equations, which is difficult to solve.

Can the locus have a closed-form solution?

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