Billy Smith, age $55,$ has been a member of the union for $30$ years, and as a result, has been

excluded from his employer's retirement plan. Billy has been offered a management position

with his firm, which will make him eligible to participate in the company$$s $401($k$)$ plan. Billy's

objective is to retire at age $65$ with $$2,000$ in monthly retirement income, exclusive of Social

Security benefits. He assumes a life expectancy of age $95.$ The union retirement plan will

provide him with $$1,000$ monthly. $($There are no matching contributions from Billy's employer

to the $401($k$)$ plan and, his income is adequate to have the required level of contributions fall

within the deferral limits of the $401($k$)$ plan. Contributions and payments, as appropriate, are

made at the beginning of each month.$)$ If the return in the company$$s $401($k$)$ plan is $10\%,$

what monthly amount will Billy have to contribute to that plan for $10$ years to meet his

objective?