
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
Michael finds another interesting aspect of quadratic polynomials and focuses on the polynomial $x^2 - sx + q$. He realizes that for certain values of $s$ and $q$, the roots $r_1$ and $r_2$ of the polynomial satisfy $r_1 + r_2 = r_1^2 + r_2^2 = r_1^{10} + r_2^{10}$. He wants to find the maximum value of $\dfrac{1}{r_1^{11}} + \dfrac{1}{r_2^{11}}$. Help Michael determine this maximum value.

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