A financial institution has the following options on Apple stock on their book.

Short $300$ call options; option delta of $0\mathrm{.}30$ and gamma of $0\mathrm{.}04($Derivative $1)$

Long $200$ put options; option delta of $-0\mathrm{.}20$ and gamma of $0\mathrm{.}04($Derivative $2)$

They are able to trade a third option with the following characteristics:

Call options; option delta of $0\mathrm{.}50$ and gamma of $0\mathrm{.}05($Derivative $3)$

They can also freely trade in Apple shares. Which of the following transactions makes the institutions book delta$-$ and gamma$-$neutral for their Apple position?

a$.$

Go long $120$ options of Derivative $3$ and go short $190$ shares of Apple stock.

b$.$

Go short $400$ options of Derivative $3$ and go short $90$ shares of Apple stock.

c$.$

Go long $80$ options of Derivative $3$ and go long $90$ shares of Apple stock.

d$.$

Go short $80$ options of Derivative $3$ and go long $90$ shares of Apple stock