Dirk Ward borrowed $$13,00000$ for investment purposes on May $16$ on a demand note proving for a variable rate of interest and payment of any accrued interest on December $31.$ He paid $$1000$ on June $10,$$$200$ on September $21.$ and $$800$ on Nowember $16.$ How much is the accrued interest on December $31$ if the rate of interest was $7\%$ on May $16,75\%$ effective August $1,$ and $78\%$ effective November $1?$

The accrued interest on December $31$ is $4$

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