The Arrow$-$Debreu model has two states and two risky assets. The initial price vector is $S=\left[\begin{array}{c}7\\ 9\end{array}\right].$

The payoff matrix is: $D=\left[\begin{array}{cc}13& 2\\ 18& 6\end{array}\right]$

$($a$)$ Find a portfolio replicating an at$-$the$-$money call option on asset $1.$

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$($b$)$ Find a portfolio replicating an at$-$the$-$money put option on asset $2.$

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$($c$)$ The investor purchased $2$ call options and $3$ put options. What portfolio replicates this combination of options?

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$($d$)$ The initial value of the portfolio from $($c$)$ is