For a given positive integer $M$, the sequence $\{a_n\}$ starts from $a_{2M+1}=M(2M+1)$ and $a_k$ is the largest multiple of $k$ no more than $a_{k+1}+M$, i.e.
$$a_k=k\left\lfloor\frac{a_{k+1}+M}{k}\right\rfloor,\quad k=1,2,\cdots,2M.$$ 
The original problem asks me to show that $a_1<4M^2$ for $M\ge 3$. Then I write a program to check large $M$s like $M\sim10000000$ and surprisingly find that
$$\lim_{M\rightarrow\infty}\frac{a_1}{M^2}=\pi.$$
Is this true and why does this happen?