
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
Consider the quadratic equation \[x^2 - tx + q = 0,\] where the roots \(a_1\) and \(a_2\) fulfill the condition \[a_1 + a_2 = a_1^2 + a_2^2 = a_1^3 + a_2^3 = \cdots = a_1^{1003} + a_2^{1003}.\] Determine the value of \[a_1^{1004} + a_2^{1004}.\]

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