
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
We want to calculate the Fibonacci sequence modulo 139.

- Verify that 12 is a solution of the equation \( y^{2} = 5 \) modulo 139.
- Deduce that the solutions of \( x^{2} - x - 1 = 0 \) modulo 139 are 64 and 76.
- Find an integer \( b \) such that \( 12b = 1 \) modulo 139.
- Deduce that if \( F_{0} = 0 \) and \( F_{1} = 1 \), then \( F_{n} \equiv b \left( 76^{n} - 64^{n} \right) \) modulo 139.

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