A corporation has a taxable income of $$8,460,000.$ At this income level, the federal income tax rate is $50\%,$ the state tax rate is $20\%,$ and the local tax rate is $10\%.$ If each tax rate is applied to the total taxable income, the resulting tax liability would be $80\%$ of taxable income. However, it is customary to deduct taxes paid to one agency before computing taxes for the other agencies. Assume that the federal taxes are based on the income that remains after the state and local taxes are deducted, and that state and local taxes are computed in a similar manner. What is the tax liability of the corporation $($as a percentage of taxable income$)$ if these deductions are taken into consideration? Construct a mathematical model. Use Gauss$-$Jordan elimination to solve the model.

The tax liability of the corporation is $\%$ of the taxable income.

$($Round to the nearest hundredth as needed.$)$