
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 \( a_{1} = 3 \), and for \( n > 1 \), let \( a_{n} \) be the largest real number such that:
\[ 4\left(a_{n-1}^{2} + a_{n}^{2}\right) = 10 a_{n-1} a_{n} - 9 \]

What is the largest positive integer less than \( a_{8} \)?

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