Next integer in this sequence, Challenge

[RLBrown]

$\sqrt{\text{mbh}_{29}}$ Challenge:

S_n = 3, 293, 7862, 32251, 7105061, 335283445, 12826573186, ?, ?, 44164106654163
S₁ through S₇ begin an infinite integer sequence, not found in OEIS.

1) Find S₈ and S₉.
2) Does S₁₀ belong to S_n?
3) If S₁₀ is incorrect, what is the correct value of S₁₀?

Motivation:

Spoiler

I have always hated the "find the next number in this sequence" type of challenge. I wanted to create a sequence who's solution is so compelling and satisfying that it would be unique. Hopefully, the hints have accomplished that goal.

 

[RLBrown]

Hint:

Spoiler

1) $\sqrt{\text{mbh}_{29}}$ encourages us to examine S_n in base 29.
2) Clicking on S₁₀ provides a tool.
3) Construct sequence R_n with R₁₀ = 44164106654163, using pattern observed in S₁ through S₇.
4) In what sense is R the additive inverse of S? Use that to complete S_n.

Think mod 29^n and look at S_n² to check your answers.
Trust me, this should be fun:)

now find S8 and S9.