II. (50 points) Given the algebraic expression $-x^{3}+100 x^{2}+x$, the letter $x$ is only allowed to take values within the set of positive integers. When the value of this algebraic expression reaches its maximum, what is the value of $x$? Prove your conclusion.
Let $x=k$, the value of the algebraic expression is $a_{k}$.
$$
a_{k+1}-a_{k}=-3 k^{2}+197 k+100 \text {. }
$$

When $00$;
When $k \geqslant 67$, $a_{k+1}-a_{k}<0$.
$\therefore$ When $x=67$, the algebraic expression reaches its maximum value.