given a 4-digit number $(abcd)_{10}$ such that both$(abcd)_{10}$and$(dcba)_{10}$ are multiples of $7$,having the same remainder modulo $37$.find $a,b,c,d$.

in a right-angled triangle $ABC$ with $\angle C=90$,$a,b,c$ are the corresponding sides.Circles $K.L$ have their centers on $a,b$ and are tangent to $b,c$;$a,c$ respectively,with radii $r,t$.find the greatest real number $p$ such that the inequality $\frac{1}{r}+\frac{1}{t}\ge p(\frac{1}{a}+\frac{1}{b})$ always holds.

$A,B,C$ are the three angles in a triangle such that $2\sin B\sin (A+B)-\cos A=1$, $2\sin C\sin (B+C)-\cos B=0$ find the three angles.

in an acute triangle $ABC$,$D$ is a point on $BC$,let $Q$ be the intersection of $AD$ and the median of $ABC$from $C$,$P$ is a point on $AD$,distinct from $Q$.the circumcircle of $CPD$ intersects $CQ$ at $C$ and $K$.prove that the circumcircle of $AKP$ passes through a fixed point differ from $A$.