
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 \( a, b, c, \) and \( d \) be complex numbers. If the set \( S = \{a, b, c, d\} \) has the property that "for any \( x, y \in S \), \( xy \in S \)", then when \( a^2=1 \), \( b^2=1 \), and \( c^2=b \), find \( b + c + d = \).

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