Bound on eigenvalues of a certain symmetric DPR1 matrix

For a set of real numbers $a_1, \dots, a_d$, define: $$ A = \begin{pmatrix} 0 & a_1 a_2 & \dots & a_1 a_n \\ a_2 a_1 & 0 & \dots & a_2 a_n \\ \vdots & & \ddots & \vdots \\ a_n a_1 & \dots & a_n a_{n-1} & 0 \end{pmatrix} $$ I would like to obtain (the tightest possible) bound on the eigenvalues of $A$. I need both a lower and an upper bound.

I know it can be expressed as a DPR1, diagonal plus rank-one, matrix, and there are algorithms for finding eigenvalues of such matrices. But I wasn't able to find a bound in terms of the numbers $a_1$, ..., $a_n$.

One set of inequalities we can get is Weyl's inequalities. Let $\lambda_1(D)\geq \cdots \geq \lambda_d(D)$ denote the eigenvalues (i.e. diagonal entries) of the diagonal matrix $D = -\operatorname{diag}(a_1^2,\dots,a_n^2)$. Let $R$ denote the rank-1 matrix with entries $r_{ij} = a_ia_j$, so that $A = D + R$. For each $i = 1,\dots,n$, we have the upper bound $$ \lambda_i(A) \leq \lambda_i(D) + \lambda_1(R) = \lambda_i(D) + \|a\|^2, $$ where $a = (a_1,\dots,a_n)$ so that $\|a\|^2 = a_1^2 + \cdots + a_n^2$. We also have the "interlacing" inequalities $$ \lambda_1(A) \geq \lambda_1(D) \geq \lambda_2(A) \geq \cdots \geq \lambda_{n-1}(D) \geq \lambda_{n}(D + A) \geq \lambda_n(D). $$