
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
1. Find the values of $i^1$, $i^2$, $i^3$, $i^4$, $i^5$, $i^6$, $i^7$, $i^8$.
2. Based on the results of (1), conjecture the pattern of values for $i^n$ ($n \in N^*$) and represent it with an equation.
3. Compute the value of $i^{2012}$.

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