A man borrows RM$250,000$ for $30$ years at $6\mathrm{.}8\%$ annually. He agrees to make annual payments of RM$15,000$ for the first $10$ years, RM$15,000+$ P for the next $10$ years, and RM$15,000+2P$ for the last $10$ years.

a$.$ Find $P.$

b$.$ Find the balance at the end of year $20.$