A stock just paid an annual dividend of $$2.$The dividend is expected to grow by $8\%$ per year for the next $3$ years. The growth rate of dividends will then fall steadily $($linearly$)$ from $8\%$ after $3$ years to $5\%$ in year $6.$The required rate of return is $12\%.$

Part $1$

What is the value of the stock if the dividend growth rate will stay $0\mathrm{.}05(5\%)$ forever after $6$ years?

Part $2$

In $6$ years, the P$/$E ratio is expected to be $23$ and the payout ratio to be $80\%.$ What is the value of the stock when using the P$/$E ratio?