Derive demand function $x(p,w)$ from utility function $u(x) = \min\{x_1, x_2\} + x_3$ 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 $x2* = \frac{w-p3x3}{p1+p2}$ and $x3* = \frac{w-(p1+p2)a}{p3}$

Confusion

How to proceed? Can I still use a Lagrangian to solve this?
Hint: Imagine that there are two coffee bars, $A$ and $B$. There is only one type of coffee in the world. My preferences are such that I always want 1 unit of sugar with 1 unit of coffee; if I consume units of coffee and sugar in the ratio $1:1$, additional units of only one of the two don't give me extra utility.

At coffee bar $A$ they sell coffee and sugar separately. That is: there is one counter for coffee, where coffee is sold for a price of $p_1$ per unit. There is another counter for sugar, where I can buy one unit of sugar for a price of $p_2$.

At coffee bar $B$ however, they only sell one unit of coffee with one unit of sugar in a bundle, the price of which is equal to $p_3$. They do not sell anything separately

At what prices do I prefer to buy my coffee with sugar at $A$, at what prices do I prefer to buy my coffee with sugar at $B$?