
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 set $A = \{(x, y) | y = 2x - 1, x \in \mathbb{N}^*\}$, and set $B = \{(x, y) | y = ax^2 - ax + a, x \in \mathbb{N}^*\}$. Is there a non-zero integer $a$ such that $A \cap B \neq \emptyset$? If it exists, find the value of $a$; if not, explain why.

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