Math Gurus - Help with Probability

[Varyon]

So, we've got a lot of people on these boards who are pretty good with more complex probability problems (or, failing that, know how to make a quick automated system to work it out for them). So, I have an exploding d6 scheme that I like more than the typical. Rather than rolling a 6 meaning you roll again and add that to 6, I have it as rolling a 6 translates into "Roll 1d+3, minimum 6," where this roll can similarly explode (becoming "1d+9, minimum 12," and so forth). This also works on the other side, where a roll of 1 becomes "1d-3, maximum 1."

The probability for 1d is easy. 2-5 are just the normal 1/6, 1 and 6 are 1/12, -4-0 and 7-11 are 1/36 each, and so forth. I have no idea how to work out the probabilities for higher numbers of dice - particularly, 3d6, so I can determine if I want to replace the standard success roll with this exploding variant, and where the criticals should be.

I actually worked this out before, going IIRC out to only 1 "explosion" (so using dice from -5 to 9, for 3d results from -15 to 27), but did so by semi-manually working out the probabilities of each result and looking at the trend. I probably made a mistake somewhere in there, and also I've lost said spreadsheet, so I can't determine how well the rules I came up with* would actually work. Thus, I'm here asking for assistance.

*For those curious, you always need at least MoS 5 for a Critical Success, and must roll a 1 or lower for skill below 10. Higher skills have a higher threshold - 2 or lower for skill 10, 3 or lower for skill 13, 4 or lower for skill 16, and 5 or lower for skill 19 - this trend doesn't continue (5 or lower is the best you can get to). Critical Failure always occurs with MoS 10, a roll of 20 or higher is a Critical Failure with MoS 5 or worse, and a roll of 24 or higher is a Critical Failure with MoF 1 or worse. Any roll of 20 is also always at least a Failure, if not a Critical Failure. There is no "always succeeds" threshold.