Selected Coefficient Sum in a Polynomial Expansion

Define the expansion of $(1-x + x^2 - x^3)^{2024}$ as $a_0 + a_1x + a_2x^2 + a_3x^3 + \dots$ where $a_n$ is the coefficient of $x^n$. Find the value of:$$a_0 + a_3 + a_6 + a_9 + \dots $$The answer is in the form $a\cdot b^c$ where $a$, $b$, $c$ are positive integers, $\gcd(a,b)=1$, and $b$ is as small as possible. Input the value of $a+b+c$.