Choice of strategic variables under relative profit maximization in asymmetric oligopoly

We consider a simple model of the choice of strategic variables under relative profit maximization by firms in an asymmetric oligopoly with differentiated substitutable goods such that there are three firms, Firm 1, 2 and 3, demand functions are linear and symmetric, marginal costs are constant, there is no fixed cost, Firm 2 and 3 have the same cost function, but Firm 1 has a different cost function. In such a model we show that there are two pure strategy sub-game perfect equilibria. One is such that all firms choose the outputs as their strategic variables, and the other is such that Firm 2 and 3 choose the outputs as their strategic variables, and Firm 1 chooses the price as its strategic variable