$($Quality of Random Number Generator$)$ We all know Fibonacci sequence,

which is defined by the following recurrence relation:

$Z_0=0,Z_1=1,Z_n=Z_{n-1}+Z_{n-2},$AAn$>1.$

This sequence was used to generate pseudo$-$random numbers in history, named the Fibonacci

generator:

$Z_i=(Z_{i-1}+Z_{i-2})modm,$

but it has serious deficiencies, as you can prove in the following two parts.

$($a$)$ Show that this generator can never produce the following arrangement of three consec$-$

utive output values: $\frac{1}{6}{U}_{i-2}.$

$(b)$ Show that the arrangement $in$ part $(a)$ should occur with probability $\frac{1}{6}$ for a "perfect"

random$-$number generator.