Recall hash puzzles from Lecture $10.$ In this question, we will see how they work in practice. This question requires programming in a language that has an implementation of SHA$-256$ hash function. You should use Python$$s hashlib for this purpose. Let

t $=2^248$

Suppose solving the hash puzzle for the current block requires you to find a nonce r$,$ an integer greater or equal to $1,$ such that

H $($str$($s$)||$ str$(2024)||$ str$($r$))<$ t

Here H is the SHA$-256$ hash function. The function $$str$$ is the string function, i$.$e$.,$ given any integer s$,$ the function str$($s$)$ casts it into a string. For example s $=2024$ becomes str$($s$)=2024.$ The symbol $||$ denotes string concatenation. Suppose s is your student ID$.$ Your strategy is to try successive values of r starting from $1,$ until the SHA$256$ of the string above falls below the target value, i$.$e$.,$ until you solve the puzzle.

$($a$)$ Implement a program that tries successive values of r$,$ i$.$e$.,$ r $=1,2,3,...,$ computes H $($str$($s$)||$ str$(2024)||$ str$($r$)),$

compares it with t and halts whenever it is less than t$,$ with the output r$.$ You need to provide the program and the output r$.$ If the value of s is different every time a new block is to be formed, what is the expected value of r before your code will find the target and why?