Three square roots under a sum constraint

Prove that for all $a, b, c \in \mathbb{R}$, where $a, b, c \geq \frac{1}{4}$ and $a + b + c = 1$, the following inequality holds: $\sqrt{4a + 1} + \sqrt{4b + 1} + \sqrt{4c + 1} \leq 5$.

Source: InternLM Lean-Workbook, record lean_workbook_419 (Apache-2.0). Complete source proposition preserved; proof developed independently.