KKT conditions for linear program where $\sum_{i=1}^{n}x_i=1$

Question

I want to find the optimal $x$ for the optimization problem: minimize $c^Tx$ subject to $x\geq0$ and $\sum_{i=1}^{n}x_i=1$?

I have calculated:

1. $c_i-\lambda_i+v=0$
2. $\sum x_i=1$
3. $\lambda_ix_i=0$
4. $x_i\geq0 , \lambda_i\geq0$

I have also found that the minimum value will be the minimum value of vector $c$, as $c^Tx\geq c_{min}$ (as stated here KKT optimisation problem ). How do i proceed to determine the optimal $x$?

Thank you in advance!