
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 \(a\) be a positive integer. The sequence \(\{u_n\}\) is defined as:
\[ u_1 = 2, \quad u_2 = a^2 + 2, \quad u_n = a u_{n-2} - u_{n-1} \quad (n \geq 3) \]

To ensure that no terms in the sequence \(\{u_n\}\) are multiples of 4, what condition must \(a\) satisfy?

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