Please show by hand and using SAGE code

$3\mathrm{.}7$ Nikita creates an RSA cryptosystem with public key

$(n,e)=(1433811615146881,329222149569169).$

In the following two problems, show the steps you take to factor $n.$

$($Don$\text{'}$t simply factor $n$ directly using a computer.$)$

$($a$)$ Somehow you discover that $d=116439879930113.$ Show how to

use the probabilistic algorithm of Section $3\mathrm{.}4\mathrm{.}3$ to factor $n.$

$($b$)$ In part $($a$)$ you found that the factors $p$ and $q$ of $n$ are very

close. Show how to use the Fermat Factorization Method of Sec$-$

tion $3\mathrm{.}4\mathrm{.}2$ to factor $n.$