
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
Suppose \(\frac{a_{1}}{b_{1}}\) and \(\frac{a_{2}}{b_{2}}\) are reduced proper fractions, and \(a_{2} b_{1} - a_{1} b_{2} = 1\). The smallest positive integer \(n\) that satisfies \(\frac{a_{1}}{b_{1}} < \frac{k}{n} < \frac{a_{2}}{b_{2}}\) is given by
$$
n = b_{1} + b_{2}, \quad k = a_{1} + a_{2}.
$$

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