6. For all real numbers $x, y$ satisfying
$$
|x|+|y| \leqslant 1
$$

the inequality
$$
\left|2 x-3 y+\frac{3}{2}\right|+|y-1|+|2 y-x-3| \leqslant a
$$

always holds. Then the minimum value of the real number $a$ is . $\qquad$
6. $\frac{23}{2}$