
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
Consider integers \( a \) and \( b \) such that \( a \neq b, a^3 - b^3 = 27x^3 \), and \( a - b = 2x \). Find the possible values of \( a \) in terms of \( x \).
A) \( a = \frac{7x + 5\sqrt{6}x}{6} \)
B) \( a = \frac{7x - 5\sqrt{6}x}{6} \)
C) \( a = \frac{7x + 5\sqrt{6}x}{6} \) or \( a = \frac{7x - 5\sqrt{6}x}{6} \)
D) \( a = 3x \) or \( a = -2x \)
E) \( a = 2x \)

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