A discrete mathematics problem by D G

Let \( a_\beta(n) \) and \(b_\beta(n) \) equal the number of digits and the number of unique digits in the base \(\beta\) representation of \(n\), respectively.

Let \( f_\beta(n) = a_\beta(n) + b_\beta(n) \).

Let \( r_\beta(g) \) equal the smallest positive integer \(n\) such that \(f_\beta(n) = g\).

Given \(\beta\), it can be shown that for all integers \( g \) above some value, \( r_\beta(g) = \beta^{g - \kappa_{\beta}} + \lambda_{\beta} \), where \(\kappa_{\beta}\) and \(\lambda_{\beta}\) are constants.

Find the sum of the digits of \((\lambda_{100})_{100}\) (\(\lambda_{100}\) represented in base \(100\)).

Examples

\( a_{10}(12321) = 5, b_{10}(12321) = 3 \)

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