$(10$ points$)$ A hypothetical equilibrium phase diagram between two elements A and B shows the following features:

$($i$)$ Element A has three solid allotropic forms with transformation temperatures of $800C$ and $1,150C$ and melts at $1,980C.$ These form solid solutions $,,$ and $$ containing element $B,$ being the low$-$temperature one.

$($ii$)$ An intermediate compound, $(A_2{B}_3)$ form melts at $1,230C.$ It has a limited solid solution for element A $,$ forming solid solution $$ and no solid solubility for element B $.$

$($iii$)$ Element B melts at $800C$ and has negligible solid solubility for A$.$

$($iv$)$ Eutectic reactions: $($a$)$ at $1000C,L(55wt.\%B)(25wt.\%B)+(60wt.\%B),$ and $($b$)$ at $650C,L(90wt.\%B)(A_2{B}_3)+B.$

$($v$)$ Peritectic reaction at $.$

$($vi$)$ Eutectoid reaction at $600C$:$(12wt.\%B)(5wt.\%B)+(65wt.\%B).$

$($vii$)$ Peritectoid reaction at $300C$:$(3wt.\%B)+(69wt\%B.)(40wt.\%B).$

$($viii$)$ At $0C$ the solubilities of B in A and A in $A_2{B}_3$ are negligible and the phase extends from $35wt.\%$ to $45wt.$

Sketch the equilibrium phase diagram. Label all of the phase fields. Assume that all phase boundaries are straight lines.