Lucia Gabrielli purchased an n$-$year par$-$value $$2,000$ bond that had a coupon rate of $9\%$ convertible quarterly; here $4$n is an integer. Her sister Elana purchased a par$-$value bond with an identical coupon rate but having a term of $2$n years. The coupons that Lucia and Elana received for the n years they both held their bonds were identical, and the yield rate used to determine the prices of each of the bonds was a nominal rate of $4\%$ convertible quarterly. Elana paid $$233\mathrm{.}01$ more than Lucia. Calculate the number of coupons in Lucia's bond; use the premium$-$discount equation. $($Round your answer to the nearest integer.$)$

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