Suppose a company has fixed costs of $$47,600$ and variable cost per unit of $\frac{1}{3}x+444$ dollars, where $x$ is the total number of units produced. Suppose further that the selling price of its product is $2172-\frac{2}{3}x$ dollars per unit.

$($a$)$ Find the break$-$even points. $($Enter your answers as a comma$-$separated list.$)$

$x=$

$($b$)$ Find the maximum revenue. $($Round your answer to the nearest cent.$)$

$

$($c$)$ Form the profit function $P(x)$ from the cost and revenue functions.

$P(x)=$

Find maximum profit.

$

$($d$)$ What price will maximize the profit? $($Round your answer to the nearest cent.$)$

$