Valley National Bank has $$1$ million in funds to be allocated between home loans, personal loans and auto loans. The annual rates of return for the loans are $7\%$ for home loans, $12\%$ for personal loans and $9\%$ for auto loans. The bank's planning committee has decided that at least $40\%$ of the funds be allocated to home loans. In addition, the planning committee has determined that the amount allocated to personal loans cannot exceed $60\%$ of the amount allocated to auto loans.