
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
In a physics problem solving session, Julia encounters a quadratic \(x^2 - tx + q\) with roots \(u_1\) and \(u_2\). She discovers that for some unknown reason, \(u_1+u_2 = u_1^2+u_2^2 = u_1^4+u_2^4\). Intrigued, she decides to calculate the maximum possible value of \(\dfrac{1}{u_1^{2009}} + \dfrac{1}{u_2^{2009}}\).

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