
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
Let \(a^2 = \frac{9}{27}\) and \(b^2 = \frac{(3+\sqrt{8})^2}{12}\), where \(a\) is a negative real number and \(b\) is a positive real number. If \((a + b)^4\) can be expressed in the simplified form \(\frac{x \sqrt{y}}{z}\) where \(x\), \(y\), and \(z\) are positive integers, what is the value of the sum \(x + y + z\)?

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