
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
For any positive integer \( n \), define the function \(\mu(n)\):
\[ 
\mu(1) = 1,
\]
and when \( n = p_{1}^{\alpha_{1}} p_{2}^{\alpha_{2}} \cdots p_{t}^{\alpha_{4}} \geqslant 2 \),
\[
\mu(n) = \begin{cases} 
(-1)^{t}, & \text{if } \alpha_{1} = \alpha_{2} = \cdots = \alpha_{t} = 1; \\ 
0, & \text{otherwise,}
\end{cases}
\]
where \( t \geqslant 1 \) and \( p_{1}, p_{2}, \cdots, p_{t} \) are distinct prime numbers. If \( A = \{x_{1}, x_{2}, \cdots, x_{k}\} \) denotes the set of all distinct positive divisors of 12, then calculate \(\sum_{i=1}^{k} \mu(x_{i})\).

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