
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
By decomposing the number \( \frac{a}{b} \) into a continued fraction, solve the equations \( a x - b y = 1 \) in integers, where
a) \( a = 101 \), \( b = 13 \)
b) \( a = 79 \), \( b = 19 \).

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