A quadratic bound under two ratio constraints

Let $a$ and $b$ be two positive real numbers with $a \le 3b \le 5a .$ Prove that $a^2+ b^2 \le \frac{10}{3} ab$.

Source: InternLM Lean-Workbook, record lean_workbook_10948 (Apache-2.0). The complete source proposition and explicit variable declarations are preserved.