GAIC Math 2024 Problem 79

Non-negative real numbers $\mathrm{x}, \mathrm{y}, \mathrm{z}$ satisfy $4 x^{2}+4 y^{2}+z^{2}+2 z=3$, then the minimum of $5 \mathrm{x}+4 \mathrm{y}+3 \mathrm{z}$ is _____.