Nash equilibrium for Bertrand Model with Spatial Differentiation

Consider a town with consumers represented by a closed interval $[0,2]$ with the consumers spread continuously and uniformly. There are two stores, $A$ and $B$ who sell the same product at $p_A$ and $p_B$ at no cost.

A consumer has a gross utility of 4 from having the product which is reduced by price paid and travel costs, which is determined as distance traveled. Each consumer only buys 1 unit of the product, and if the consumer does not buy from either store, he has a utility of 0. For example, if a consumer is located at $x=1.5$ and buys from store B at $p_B=1$ which is located at $x_B=2$, then his utility is $4-1-|2-1.5| = 2.5$.

Suppose that $A$ is located at $x_A=0$ and $B$ is located at $x_B=2$. Also, suppose stores much charge at $p le 4$.

Suppose stores are profit maximizers. What are the equilibrium prices, quantities, and profits for both stores?

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My initial thoughts are that, in this scenario, the profit functions for each store is denoted as:

$$ pi_A=p_A+p_A(frac{p_B-p_A}{2}) $$

$$ pi_B=p_B-p_B(frac{p_B-p_A}{2}) $$

We can take the first order condition for each, to get the maximizing profit.

$$ frac{partial pi_A}{p_A} = 1 + frac{p_B}{2} – p_A = 0 implies p_A = frac{p_B}{2} + 1 $$

$$ frac{partial pi_B}{p_B} = 1 – p_B + frac{p_A}{2} = 0 implies p_B = 1 + frac{p_A}{2} $$

By substituting one into the other, we have that $p_A=p_B=2$, $q_A=q_B=1$ and $pi_A=pi_B=2$.

I’m not very sure about this solution or the direction I took to solve it. Any advice here would be appreciated on what other direction I should try.