The picture below shows a beam$-$column joint. The flexural steel bars of this beam are all D$-29$ steel bars, and the yield strength is $4200kg\frac{f}{c}{m}^2.$ The beam ribs are made of D$10$ steel bars with a yield strength of $2800kg\frac{f}{c}{m}^2.$ From the support point, four beam ends are arranged with a spacing of $8$ cm $,$ and the subsequent spacing is $14$ cm $.$ Using constant$-$weight concrete with a strength of $280kg\frac{f}{c}{m}^2,$ all steel bars are uncoated, and non$-$seismic structural materials, ask for:

$($a$)$ If the area of the positive moment reinforcement required for this beam is $12$ cm $2,$ try to calculate the minimum extension length of the positive moment reinforcement using a simplified estimation method. $(10\%)$

$($b$)$ If the area of the negative moment reinforcement required for this beam is $11\mathrm{.}5c{m}^2,$ try to calculate the minimum extension length of the negative moment reinforcement by the detailed calculation method. $(15\%)$