AutoDrive Inc., a company specializing in autonomous driving technology, is the target of a leveraged buyout by Alpha Investments, a private equity firm. Alpha is acquiring AutoDrive on a cash$-$free, debt$-$free basis for a total enterprise value of $10\mathrm{.}0$x the company$$s $2023$ projected EBITDA of $$90$ million. Alpha plans to finance the acquisition with $$500$ million of new debt and new equity from funds managed by the firm. The existing management team of AutoDrive is rolling over a portion of their equity proceeds into the deal so that they have $25\%$ pro forma ownership.

Assuming that AutoDrive has no existing debt and no excess cash on its balance sheet at close, and $$10$ million of transaction$-$related expenses, what is the size of the equity check that Alpha Investments needs to write to finance this LBO transaction?

A$)$$$102\mathrm{.}5$ million

B$)$$$150\mathrm{.}0$ million

C$)$$$300\mathrm{.}0$ million

D$)$$$307\mathrm{.}5$ million