
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given the equation $ax^2+bx+c=0$ ($a \neq 0$) has two roots $x_1$ and $x_2$. Let $s_1 = x_1^{2005} + x_2^{2005}$, $s_2 = x_1^{2004} + x_2^{2004}$, and $s_3 = x_1^{2003} + x_2^{2003}$. Find the value of $as_1 + bs_2 + cs_3$.

Present the answer in LaTex format: \boxed{Your answer}
user