Consider a $1n$ strip of cells containing white, grey, and black stones with the following rules:

$($i$)$ Every cell contains exactly one stone.

$($ii$)$ There is at least one black stone on the strip.

$($iii$)$ A grey stone cannot be adjacent to a black stone.

$($iv$)$ All cells to the left of a white stone are occupied by white stones.

For each $i=1,2,$dots,$n,$ let $w_i,g_i,b_i$ be boolean variables. We interpret $w_i=1$ to mean there is a white stone on cell i and $w_i=0$ to mean there is not a white stone on cell $i.$ Similarly for $g_i($grey$)$ and $b_i($black$)$

Write down a boolean formula $F$ such that for any assignment of values $(0$ or $1)$ to the variables, $F$ evaltatates to true if and only if the rules above are respected.

You may use the following notations, similar to the sigma notation used for sums:

$)=(1$