Make the constraint relation between $a_{1,}{a}_2$ and $a_3$.

 Points 1, 2, 3 and 4 are movable. Let their displacements from the fixed dotted line be $x_1,x_2,x_3$ and $x_4$ (ignoring the length over the pulley)

We have,

$x_1+x_4=I_1$ (length of first string)

and

$\left(x_2-x_4\right)+\left(x_3-x_4\right)=l_2$

or

$x_2+x_3-2x_4=l_2$

On double differentiating with respect to time, we get

and

$$\begin{gathered} a_1+a_4=0 \\ {a}_2+a_3-2a_4=0 \end{gathered}$$

$a_4=-a_1$

But

We have, $a_2+a_3+2a_1=0$

This is the required constraint relation between $a_{1, }{a}_{2 }and a_3$.