Analytical Geometry!

Algebra Level 5

Draw a tangent line of parabola \(y=x^2\) at the point \(A(1,1)\). Suppose the line intersects the \(x\)-axis and \(y\)-axis at \(D\) and \(B\) respectively. Let point \(C\) be on the parabola and point \(E\) on \(AC\) such that \(\dfrac{AE}{EC}=\lambda_{1}\). Let point \(F\) be on \(BC\) such that \(\dfrac{BF}{FC}=\lambda_{2}\) and \(\lambda_{1}+\lambda_{2}=1\). Assume that \(CD\) intersects \(EF\) at point \(P\). When point \(C\) moves along the parabola, the equation of the trail of \(P\) can be expressed in the form \(\dfrac{1}{a}(mx+b)^2\). Where \(gcd(m, b)=1\), find the value of \(a+b+m\).

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