
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
Find all ordered 4-tuples of integers \((a, b, c, d)\) (not necessarily distinct) satisfying the following system of equations:
\[
\begin{aligned}
a^2 - b^2 - c^2 - d^2 &= c - b - 2, \\
2ab &= a - d - 32, \\
2ac &= 28 - a - d, \\
2ad &= b + c + 31.
\end{aligned}
\]

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