
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 $z = (x-a)^3 + (x-b)^3$ where $a$ and $b$ are constants. For what value of $x$ is $z$ a minimum?
**A)** $\frac{a+b}{2}$  
**B)** $a+b$  
**C)** $\sqrt{ab}$
**D)** $\sqrt{\frac{a^2+b^2}{2}}$  
**E)** $\frac{a+b}{2ab}$

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