Dirk Ward borrowed $$13,000\mathrm{.}00$ for investment purposes on May $13$ on a demand note providing for a variable rate of interest and payment of any accrued interest on December $31.$ He paid $$900$ on June $8,$$$200$ on September $10,$ and $$600$ on November $20.$ How much is the accrued interest on December $31$ if the rate of interest was $8\%$ on May $13,8\mathrm{.}6\%$ effective August $1,$ and $9\mathrm{.}2\%$ effective November $1?$

The accrued interest on December $31$ is $

$($Round the final answer to the nearest cent as needed. Round all intermediate values to six decimal places as needed.$)$