variation of Vitali in $mathbb{R}^2$

Let $B(x, r)$, be a two dimensional open disc with center $x in mathbb{R}^2$ and radius $r > 0$. Consider a set $E subset mathbb{R}^2$ such that $E subset bigcup_{i = 1}^n B(x_i, r_i)$ and $sum_{i = 1}^n r_i leq 1$ Show that then there exists a collection of non-overlapping open discs $B(y_i, R_i)$ such that $E subset bigcup_{i = 1}^n B(y_i, R_i)$ and $sum_{i = 1}^n R_i leq 1$

Clearly if the balls $B(x_i, r_i)$ are non overlapping then we are done. If not then at least $2$ balls must overlap.

First attempt
Let $B(x_1,r_1)$ be the ball whose radius is the smallest such that $B(x_1,r_1)$ overlaps with at least one other ball. Pick $B(x_j,r_j)$ whose radius is maximal among all the balls overlapping with $B(x_1,r_1)$, and note that the sums of the radius of the balls in $B(x_{j},r_{j} + r_1) bigcup_{i = 2, i neq j}^n B(x_i,r_i)$ is the same as the sum of the radius of the balls in $bigcup_{i = 1}^n B(x_i, r_i)$. The problem here is that $B(x_1,r_1)$ is not necessarily contained in $B(x_j,r_j + r_1)$.

Second attempt Let $B(x_1,r_1)$ be the ball whose radius is the smallest such that $B(x_1,r_1)$ overlaps with at least one other ball. Pick $B(x_j,r_j)$ whose radius is maximal among all the balls overlapping with $B(x_1,r_1)$, and note that $B(x_1, r_1)$ is contained in $B(x_j, r_j + 2r_1)$. The problem here is that the sums of the radius of the balls in $B(x_{j},r_{j} + 2r_1) bigcup_{i = 2, i neq j}^n B(x_i,r_i)$ is Not the same as the sum of the radius of the balls in $bigcup_{i = 1}^n B(x_i, r_i)$ and therefore not necessarily less then or equal to $1$.

Since I want to cover $E$ and have radius that sum up to be less then or equal to $1$, I know I want to maximize surface area while keeping the sum of the radi as small as possible so it dose feel logical to get rid of smaller circles while expanding upon larger once. Any help would be appreciated.

Edit
So after thinking about it a little longer I realized it might be impossible to create the disjoint collection of balls $B(y_i,R_i)$ such that each $y_i = x_i$ and I might have to move the discs. For example if $E$ was the union of $2$ balls of each radius $1/2$, one centered at the origin and one centered at $(7/8,0)$ then there is no way to just expand one ball without moving it so that its radius remains $1$ but it covers the other ball.

So now I am thinking start with letting $B(x_1,r_1)$ be the ball whose radius is the smallest such that $B(x_1,r_1)$ overlaps with at least one other ball. Pick $B(x_j,r_j)$ whose radius is maximal among all the balls overlapping with $B(x_1,r_1)$, Consider the ball $B( ?,r_1 + r_j)$ so I need to find a suitable value for $?$. I am temped to average the two centers $x_1, x_j$ but I know it should be closer to $x_j$, but then that makes this really complicated.