$$\begin{aligned} & \mathrm{P} \equiv\left(\frac{\alpha-2}{2}, \frac{\beta-1}{2}\right) \equiv\left(\frac{\gamma+1}{2}, \frac{\delta}{2}\right) \\ & \frac{\alpha-2}{2}=\frac{\gamma+1}{2} \text { and } \frac{\beta-1}{2}=\frac{\delta}{2} \\ & \Rightarrow \alpha-\gamma=3 \ldots .(1), \beta-\delta=1 \ldots \ldots (2) \end{aligned}$$

Also, $$(\gamma, \delta)$$ lies on $$3 x-2 y=6$$

$$3 \gamma-2 \delta=6$$ ..... (3)

and $$(\alpha, \beta)$$ lies on $$2 x-y=5$$

$$\Rightarrow 2 \alpha-\beta=5 \text {. }$$

Solving (1), (2), (3), (4)

$$\begin{aligned} & \alpha=-3, \beta=-11, \gamma=-6, \delta=-12 \\ & |\alpha+\beta+\gamma+\delta|=32 \end{aligned}$$