Greedy Algorithm for an Abelian Group

Question

In Gallian's Contemporary Abstract Algebra, he mentioned about this Greedy Algorithm for an Abelian group. It's basically a procedure of expressing a finite Abelian group as an internal direct product. But the validity of the algorithm is not clear.

I included the algorithm here:

Greedy Algorithm for an Abelian Group of Order $p^n$ ($p$ is prime)

1. Compute the orders of the elements of the group $G$.
2. Select an element $a_1$ of maximum order and define $G_1=\langle a_1\rangle$.
3. If $|G_1|=|G|$, stop. Otherwise, replace $i$ by $i+1$.
4. Select an element $a_i$ of maximum order $p^k$ such that $p^k\leq|G|/|G_i|$ and none of $a_i$, $a_i^p$, $a_i^{p^2}$,..., $a_i^{p^{k-1}}$ is in $G_i$ and define $G_i=G_{i-1}\times\langle a_i\rangle$.
5. Return to Step 3.

Internal direct product requires that none of the elements of $\langle a_i\rangle$ except identity should be in $G_{i-1}$ so why the condition in step 4 is weaker?