$10$ points$]$ Suppose

$$

M is a single$-$tape TM with tape alphabet

$\backslash$Gamma

$\backslash$Gamma and state space

$$

Q $($where

$$

$\backslash$Gamma

$$

$=$

$10$

$\backslash$Gamma $=10,$

$$

$$

$$

$=$

$10$

$$Q$=10).$ On an input

$$

w of length

$$

n$,$ suppose we are guaranteed that the TM on input

$$

w never goes beyond the first

$2$

$$

$2$n cells. Define

$$

$=$

$100$

$$

$$

$40$

$$

T$=100$n$40$n$.$ Prove that if the TM

$$

M on input

$$

w does not halt in the first

$$

T steps, then it will never halt on input

$$

w$.$