I know how to solve the two-good case with u(x) = {min x1, x2}, but the addition of x3 confuses me.

**Problem: derive the demand function x(p,w) from $u(x)$ = min{x1, x2} + x3**

**What I did so far**
We assume that in optimum $x1 = x2$.

Set up the budget constraint $p1x1 + p2x2 + p3x3 = w$

Rewrite budget constraint as $(p1+p2)x1 + p3x3 = w$ or $(p1+p2)x2 + p3x3 = w$

We can write $x1*$ or $x2* = (w-p3x3)/(p1+p2)$ and $x3* = (w-(p1+p2)a)/(p3)$

**Confusion**

How to proceed? Can I still use a Lagrangian to solve this?