Lindsey wants to accumulate $$31,000$ in a fund with payments of $$2,000$ made at the end of every quarter. The interest rate is $6\mathrm{.}58\%$ compounded quarterly.

a$.$ How many payments $($rounded up to the next payment$)$ will it take to accumulate this amount?

$20$ payments

$12$ payments

$13$ payments

$14$ payments

b$.$ How long will it take to accumulate this amount?

$5$ years and $6$ months

$5$ years and $4$ months

$6$ years and $6$ months

$3$ years and $6$ months