Consider the language

$L_3=\{w_1$#$w_2|w_{1}in\{0,1{\}}^{*}$ and $w_{2}in\{0,1{\}}^{*}$ such that $[w_2]=3[w_1]$ and $|w_1|=|w_2|\}.$

Here, for a string $w,[w]$ represents the value of the string with first bit as the least significant bit.

For example $[0011]=12,[101]=[1010]=5,$ and $[001]=4.$ Therefore, $0010$#$0011in{L}_3.$ Give

a single tape turing machine that recognizes $L_3.$ Precise high level description of Turing ma$-$

chine suffices.

I have seen this question answered on Chegg once already, but the answer was confusing and frankly didn't make any sense.