You are given an array of m lamp posts with non$-$negative integer light values, where $0$ means that the lamp post can only light up a section outside one house, $1$ means it can light a given house plus the two immediately neighboring houses on either side of this house, and a light value of k means it can light a given house plus the next k houses on either side of this house $(2$k $+1$ consecutive houses in total$).$ You can place a lamp post next to any house. A house may be lit by several lamp posts.

$($a$)$If we have an array of five lamp posts $[3,5,2,6,1],$ what is the minimum number of lamp posts needed to light the street with len $=24$ houses?

$($b$)$Prove the greedy algorithm that sorts the set of lamp posts in decreasing order $($and picks lamp posts until all houses on the street are lit$)$ is optimal.