Lindsay Construction has an opportunity to build five Recreation Centers during the next year. The expected profit and expected cost for each of the Recreation Centers are shown in the table below.

Recreation Centers

Expected Net Profit $($$$1000)$

Expected Cost $($$$1000)$

$1$

$340$

$980$

$2$

$160$

$810$

$3$

$240$

$650$

$4$

$175$

$850$

$5$

$210$

$1100$

The company has budgeted $$3,600,000$ for the construction of Recreation Centers during the next year. Also, due to various legal restrictions and marketing considerations, the following relationships among the projects must be met.

At least one of the four Recreation Centers $1,3,$ and $4,$ and $5$ must be constructed.

If Recreation Centers $1$ is constructed, then Recreation Centers $3$ also must be constructed.

Lindsay Construction formulates a Binary Integer Programming to determine which shopping malls they should build in order to maximize total net profit. Binary decision variables X$1,$ X$2,$ X$3,$ X$4,$ and X$5$ are defined with value $1$ representing the decision to construct the corresponding shopping mall and value $0$ otherwise. Which of the following models represents the mathematical formulation of$$this$$problem$?$