Alternating Coefficients

Algebra Level 4

Let $P(x)=(x^{3}-x^{2}+x+4)^{2001}=a_{0}+a_{1}x+a_{2}x^{2}+....+a_{6003}x^{6003}$ .

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