
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 \(a_0 = 3\), \(b_0 = 4\), and define the sequences \(a_{n+1}\) and \(b_{n+1}\) by
\[a_{n+1} = \frac{a_n^2}{b_n}, \quad b_{n+1} = \frac{b_n^2}{a_n}\]
for all \(n \geq 0\). Determine \(b_7\) in the form \(\frac{4^m}{3^n}\) where \(m\) and \(n\) are integers, and find the ordered pair \( (m,n) \).

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