
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 \( a_{1}, a_{2}, a_{3}, a_{4} \) be four distinct numbers from the set \(\{1, 2, \ldots, 100\}\) such that they satisfy the following equation:
\[ 
\left(a_{1}^{2}+a_{2}^{2}+a_{3}^{2}\right)\left(a_{2}^{2}+a_{3}^{2}+a_{4}^{2}\right) = \left(a_{1} a_{2}+a_{2} a_{3}+a_{3} a_{4}\right)^{2}.
\]

How many such ordered quadruples \(\left(a_{1}, a_{2}, a_{3}, a_{4}\right)\) are there?

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