d$)$ A financial institution has the following portfolio of over$-$the$-$counter options on gold:

$\backslash$table$[[$Type$,$Position,$\backslash$table$[[$Delta of$],[$option$]],\backslash$table$[[$Gamma of$],[$option$]],\backslash$table$[[$Vega of$],[$option$]]],[$Call$,500,0\mathrm{.}5,0\mathrm{.}6,0\mathrm{.}9],[$Call$,-420,0\mathrm{.}2,1\mathrm{.}2,0\mathrm{.}7],[$Put$,-290,-0\mathrm{.}8,1\mathrm{.}6,1\mathrm{.}2],[$Put$,340,-0\mathrm{.}2,1\mathrm{.}3,0\mathrm{.}6]]$

A traded option is also available with a delta of $0\mathrm{.}35,$ a gamma of $1\mathrm{.}3,$ and a vega of $1\mathrm{.}6.$

i$)$ What position in the traded option and in gold would make the portfolio both vega neutral and delta neutral?

ii$)$ A second traded option is available with delta of $0\mathrm{.}25,$ gamma of $1\mathrm{.}4,$ and vega of $1\mathrm{.}1.$ What position in the two traded options would make the portfolio delta, gamma, and vega neutral?