Brian made periodic deposits into a savings account at the end of every month for $5$ years. The investments were earning $4\mathrm{.}50\%$ compounded quarterly and grew to $$12,750\mathrm{.}00$ at the end of $5$ years.

a$.$ Calculate the size of the month$-$end deposits.

$$193\mathrm{.}56$

$$189\mathrm{.}97$

$$149\mathrm{.}94$

$$186\mathrm{.}49$

b$.$ How long will it take for the $$12,750\mathrm{.}00$ to accumulate to $$34,225\mathrm{.}00$ if the interest rate remained the same and he continued making the same month$-$end deposits throughout the term?

$7$ years and $7$ months

$11$ years and $7$ months

$6$ years and $7$ months

$6$ years and $9$ months