Lookback put on a two-step tree

A stock starts at $S_0 = 100$ and each step moves up by a factor $u = 1.5$ or down by $d = 0.5$. Cash grows by $1 + r = 1.1$ per step. A floating-strike lookback put pays $M_2 - S_2$ at time 2, where $M_2 = \max(S_0, S_1, S_2)$.

(a) Find the risk-neutral up-probability.

(b) Price the lookback at time 0.

(c) Explain why the stock price alone is not enough to price this option by backward induction, and what state you would use instead.