MHB Next integer in this sequence, Challenge

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

Sn = 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 Sn?
3) If S10 is incorrect, what is the correct value of S10?

Motivation:
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.
 
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Hint:

1) $\sqrt{\text{mbh}_{29}}$ encourages us to examine Sn in base 29.
2) Clicking on S10 provides a tool.
3) Construct sequence Rn 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 Sn.

Think mod 29^n and look at Sn2 to check your answers.
Trust me, this should be fun:)
now find S8 and S9.
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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