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294. Sum of digits - experience #23

For a positive integer k, define d(k) as the sum of the digits of k in its usual decimal representation. Thus d(42) = 4+2 = 6.

For a positive integer n, define S(n) as the number of positive integers k < 10ⁿ with the following properties :

• k is divisible by 23 and
• d(k) = 23.

You are given that S(9) = 263626 and S(42) = 6377168878570056.

Find S(11¹²) and give your answer mod 10⁹.