Let three real numbers \( a, b, c \) be in arithmetic progression and \( a + 1, b, c + 3 \) be in geometric progression. If \( a > 10 \) and the arithmetic mean of \( a, b \) and \( c \) is \( 8 \), then the cube of the geometric mean of \( a, b \) and \( c \) is:
