$(10)$ Mr$.$ Valdez has $$10,000$ to invest at time $t=0,$ and three ways to invest it$.$ Investment account I is governed by compound interest with an annual effective discount rate of $3\%.$ Investment account II has force of interest equal to $\frac{\mathrm{.}04}{1+\mathrm{.}05t^2}.$ Investment account III is governed by the accumulation function $a^{\text{I}\text{I}\text{I}}(t)=(1-\mathrm{.}005t^2{)}^{-1}.$ Mr$.$ Valdez can transfer his money between the three investments at any time. What is the maximum amount he can accumulate at time $t=5?[$HINT: At all times, Mr$.$ Valdez wishes to have his money in the account that has the greatest force of interest at that moment. Therefore, begin by determining the force of interest function for each of the investment accounts. Next decide for which time interval Mr$.$ Valdez should have his money in each of the accounts. Assume that he accordingly moves his money to maximize his return. You will then need the accumulation functions for the accounts in order to determine Mr$.$ Valdez's balance at $t=5.$ Remember to use Important Fact $(1\mathrm{.}7\mathrm{.}4).]$

Answer: $$12,140\mathrm{.}26$