RB_CH02_Q036. American out call with up and down barriers

Consider an American-style double-barrier 'out-in' call option. There is an out barrier above the current stock price (an 'up barrier') and an in barrier below the current stock price (a 'down barrier'). This option has a payoff only if the following events happen: first, the stock price path includes a fall in price below the down barrier (i.e., the option is 'knocked in'); second, the stock price path does not include a rise in price above the up barrier (i.e., the option is not 'knocked out'); and third, the option is exercised when the stock price is above the strike (i.e., the option is in-the-money at exercise). This option is both path-dependent and American-style. Is there an easy technique for valuing this option?