If $u_1,…,u_n$ are in $\mathbb R^n$ and the set $(u_i+e_i)$ is LD, show that $\sum_{i=0}^n ||u_i|| \ge 1$

The exercise is to show that $ \sum_{i=0}^n ||u_i|| \ge 1$ , knowing that the set of $ (u_i+e_i)$ in $ \mathbb R^n$ is LD, where $ e_i$ is the canonical basis.

I know that $ e_i$ are ortogonal vectors of modulus $ 1$ , and that being LD means that at least one of the vectors $ (u_i+e_i)$ can be expressed by a linear combination of the others. From this I tried isolating $ e_i$ , and thinking of inner product to use something like Cauchy-Schwarz to get an inequality, but was unable to develop the solution.