
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 \( a_0 = 1 \), \( b_0 = 5 \), and
\[
a_{n + 1} = \frac{a_n^2 + 1}{b_n}, \quad b_{n + 1} = \frac{b_n^2 + 1}{a_n}
\]
for all \( n \geq 0 \). Find \( b_7 \) expressed in the form \( \frac{5^m}{1^n} \) for some integers \( m \) and \( n \).

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