A simply supported beam is a single$-$reinforced rectangular beam$($Fig$\mathrm{.}2)$ with a span of $7$ m $,$ a beam width b of $40$ cm $,$ an effective depth d of $55$ cm $,$ a static load $($including its own weight$)2000kg\frac{f}{m},$ and a live load of $1800kg\frac{f}{m},fc=350kg\frac{f}{c}{m}^2,$ fy $=4200kg\frac{f}{c}{m}^2.$

$($a$)$ Please use the equivalent rectangular stress distribution method to design the cross$-$sectional area $A_s$ of the tension steel bar. $(10\%)$

$($b$)$ According to the $A_s$ obtained in the sub$-$problem $($a$),$ use four single$-$numbered steel bars to configure the tension bars with the most economical amount of tension bars. Calculate the design strength of the bending moment after the reinforcement, and check whether the amount of reinforcement of the beam meets the specifications. $(15\%)$