To provide for expansion, a company has established a sinking fund earning $10\%$ quarterly. The fund is anticipated to reach a balance of $$51,000$ in $14$ years. Payments are made at the beginning of every $3$ months.

$($a$)$ What is the size of the periodic payment?

$($b$)$ What is the accumulated balance at the end of payment period $47?$

$($a$)$ The size of the periodic payment is $

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