A trader holds the following apple stock $($AAPL$)$ options in the portfolio today:

In the table above, a position of $500$ on AAPL Option #$1$ means the trader holds $500$ of that option. In addition, the trader can also trade AAPL stock and another apple stock option: AAPL option #$5.$ This option has a delta of $0\mathrm{.}5,$ a gamma of $1\mathrm{.}2,$ and a vega of $0\mathrm{.}6.$

$($a$)$ What is estimated change in portfolio value if apple stock price goes up by $$0\mathrm{.}1?$

$($b$)$ How should the trader trade to delta$-$hedge this portfolio today?

$($c$)$ What position in the AAPL option #$5$ and AAPL stock would make the portfolio both gamma neutral and delta neutral today?

$($d$)$ Please briefly explain why we cannot make the portfolio delta, gamma, and vega neutral with only one addition option $($AAPL option#$5)?$

$($e$)$ Further suppose there is another AAPL option #$6.$ This option has a delta of $0\mathrm{.}2,$ gamma of $0\mathrm{.}5,$ and vega of $0\mathrm{.}7.$ What position in the AAPL option #$5,$ AAPL option #$6,$ and AAPL stock would make the portfolio delta, gamma, and vega neutral today?