
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
Determine the number of ordered triples \((x, y, z)\) of integers that satisfy the following system of equations:
\[\begin{array}{l}
x^2 - 2xy + 3y^2 - 2z^2 = 25, \\
-x^2 + 4yz + 3z^2 = 55, \\
x^2 + 3xy - y^2 + 7z^2 = 130.
\end{array}\]
A) 0
B) 1
C) 2
D) a finite number greater than 2
E) infinitely many

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