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With what you described, you did not bust with three 3's at first, then decided to keep rolling. Whatever you roll next is looked at by itself, you cannot add additional 3's to the three 3's you set aside previously.
It's mostly a matter of risk management - you either gamble and hope to find a 1, 5 or three of a kind (at least) with the die you have left, or play it safe, score your points, and pass to your opponent.
And where do it say it MUST be a 1 and / or a 5 on the fist throw?
https://en.wikipedia.org/wiki/Farkle
See the standard scoring section.
You cannot add additional 3's to the 3's you have already collected; the only way to get four of a kind or higher is if you were to get them with a single dice roll.
"Each scoring combination must be achieved in a single throw.[5] For example, if a player has already set aside two individual Dice value 1´s and then throws a third with the four dice remaining, they do not have a triplet of Dice value 1´s for a score of 1000 but merely three individual Dice value 1´s for a score of 300."
EDIT: To add, the above example only allows adding the one 1 from the second dice throw since you can score it "by itself". In your example, you can score the three 3's, but the singular 3 you got in your second throw cannot be scored by itself; only if you had all 3's with your three remaining dice, or I THINK if you had a small straight. IIRC I scored a straight before when playing the game ingame, only did so twice though so far haha
1s and 5s score points, as does any combination of 3 or more of the same dice.
If you have none of the above in the throw, you're busted.
The dice you've already set aside to score a point no longer counts and are not re-inserted in the game until you pass your turn or manage to remove all six dice whilst scoring points.
1 + 2 + 3 + 4 + 5 + 6 is my favorite, I think I had it 2 times only and I played dice enough.
Actually it's in the game, you can't miss tutorial help message unless you specifically disable them.
You have 6d6, you want to focus on overall points and combos. 1 pip is worth 100, 5 pips are worth 50. 3 of a kind are however many pips are on the dice, the points stack if you have 4-6 of a kind. Getting a straight is also something to look at, so 1-6, 1-5, 2-6 and so on. Keep in mind that the rolls are "random", so I would bet very little until you are comfortable with how the game works. I would be wary of risks. If you are down to two dice from the initial roll, unless you need to make a big score, don't put yourself in a situation to bust.