Alternating Coefficients

Algebra Level 4

Let P(x)=(x3x2+x+4)2001=a0+a1x+a2x2+....+a6003x6003P(x)=(x^{3}-x^{2}+x+4)^{2001}=a_{0}+a_{1}x+a_{2}x^{2}+....+a_{6003}x^{6003}.

The sum a0+a2+a4+...+a6002a_{0}+a_{2}+a_{4}+...+a_{6002} can be expressed in the form 5b+c2\dfrac{5^{b}+c}{2}, where b and c are positive integers, and b is as large as possible. Find the value of 5+b+c+25+b+c+2.

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