Question : Consider the following bargaining problem: $20$ lakh rupees needs to be split between X and Y $.$ Y gets to make an initial offer. X then gets to respond by either accepting or rejecting Y$\text{'}$s initial offer. If X accepts, the game ends and the initial offer would be their pay$-$off. If X rejects, then the game moves to the second period where X makes an offer. In the second period, Y has two choices$-$ either accept or reject X$\text{'}$s offer. If the offer is accepted, then X$\text{'}$s offer would be the pay$-$off. Otherwise, both are getting $0.$ Y$\text{'}$s discount factor is $0\mathrm{.}8$ and X$\text{'}$s discount factor is $0\mathrm{.}9.$

$($a$)$ Represent this game in extensive form $($draw the game tree only$).(2$ points$)$

$($b$)$ Find subgame perfect equilibrium $($SPE$)$ of this bargaining game. $(3$ points$)$