
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
Two quadratic equations with unequal leading coefficients,
$$
(a-1) x^{2} - \left(a^{2}+2\right) x + \left(a^{2}+2a\right) = 0
$$
and
$$
(b-1) x^{2} - \left(b^{2}+2\right) x + \left(b^{2}+2b\right) = 0
$$
(where \(a\) and \(b\) are positive integers), have a common root. Find the value of \(\frac{a^{b} + b^{a}}{a^{-b} + b^{-a}}\).

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