Decomposition of a sum of matrix products with itself

Question

Given tall matrix $B \in \mathbf{R}^{n \times r}$, where $n \gg r$, and vector $c \in \mathbf{R}^n$, let

$$ A A^T := B B^T + c c^T $$

I would like to get any possible matrix $A$ efficiently. If possible, smaller than a square matrix. Is there a solution? Would it help if $BB^T$ is a diagonal matrix?

Answer 1

$$A = \begin{bmatrix} B & c \end{bmatrix}$$