roblem $1(20$ points$).$ Given the sugar cane with the length of $7$ and the price table as below.

$\backslash$begin$\{$tabular$\}\{$l$|$lllllll$\}$

length & $1$ & $2$ & $3$ & $4$ & $5$ & $6$ & $7$

$\backslash$end$\{$tabular$\}$

$\backslash$begin$\{$tabular$\}\{$l$|$lllllll$\}$

price & $1$ & $5$ & $8$ & $9$ & $10$ & $17$ & $17$

$\backslash$end$\{$tabular$\}$

$($a$)$ What is the optimal strategy of sugar cane cutting $(5$ points$)?$ What is the best total price points$)?$

$($b$)$ Please provide an iterative algorithm for Cut$\_$Sugar$\_$Cane and analysis the running time points$).$