In the long run, what you would need to do is have multiple of each highest tier base tool, elemental all of them, select a set that have the best elemental benefit, use this set, upgrade their level by feeding your old tool set to them and then reforge them to the best secondary stats, in that exact order.

Having multiple of each is much more effective at coming up with a perfect alignment set than rerolling, Using a simple example of 3 of each of 2 tools, if you were doing rerolling, one will be rolled first and then the other to match it, which means 5 attempts(note total consuming 6 elemental stone, one is used on the fist tool). Whereas if you had 3 of each, there are 3x3=9 possible pairing, much more likely to have *ignment. This only gets better as it grow bigger because the combinations keep multiplying on the one side and merely add up on the other.

It doesn't matter how many combination there be, because there is a large number of acceptable combination, and they are made from very few different possible pieces.
There are only 20 possible state for corner pieces, and 60 possible state for non-corner pieces. If you have one of every state of every piece, you will be able to make all the combination.

The question is then, how much does it cost to get every single state?
That is the coupon collector's problem.
The expected amount of roll required to get all 20 state of something is 72 (see coupon collector's problem) while for 60 is 281. Therefore, the expected number of stones you need to have the full set is 1974.

But the fact of the matter is, even for an optional set where, for example, you want 5 attack, 5 nin and 3 other connections, you do not need to collect them all.
Fact of the matter is, for this combo, for example, there are only a few "important" combination--ones where both attack and nin show up on the stone. There are a large number of state that you simply don't care for. As such, while the important state can end up as the last one, it's not expected to be so for all of them. Chances are, you can find an optimal set with under 300-500 stones, provided that you keep every single piece that can be a potential solution (since there are, I believe, 270 solutions)

Long story short, because you won't reforge just once.
Now, you can do your first reforge earlier, but in the long run, you will be doing likely 5 reforge per gear, and each subsequent reforge will be less and less meaningful. This is just like leveling, where the first time it eat 6 orange gear, it's going to gain half a dozen levels, but with each additional set eaten, it grows less and less. You can intermingle them but the final reforge will come last.

Besides, you can't START reforging until you collect a decent collection of them anyway. It would be a sad day for you to discover that NONE of the stat on your reforged gear was at the maximal... so at least 20, preferably 50 samples are needed before making your decision (have fun with your excel charts, I dunno about you, I keep one). Otherwise, instead of 5 reforges, you might end up with 10...

I see, on average, I believe more gold stone in the choices... but I have horrid luck at picking them.
I've gotten two in the last 20 or so runs, when I see on average at least one per run (note that while there are some with none, there are others with 3 to even it out), I hope this is just luck, which will even out in the long run as opposed to... well... you know...

I also missed red gourd six times.... that's annoying too...