
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
Given complex numbers $a$, $b$, $c$, and $d$, if the set $S=\{a, b, c, d\}$ has the property "for any $x, y \in S$, it must be that $xy \in S$", when $$\begin{cases} a=1 \\ b^{2}=1 \\ c^{2}=b\end{cases}$$, find the value of $b+c+d$.

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