
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
Given positive integers \(a\) and \(b\) such that \(a > b\), define a sequence of ordered pairs \((a_l, b_l)\) for nonnegative integers \(l\) by \(a_0 = a, b_0 = b\), and \((a_{l+1}, b_{l+1}) = (b_l, a_l \bmod b_l)\), where for all positive integers \(x\) and \(y\), \(x \bmod y\) is defined to be the remainder left by \(x\) upon division by \(y\). Define \(f(a, b)\) to be the smallest positive integer \(j\) such that \(b_j = 0\). Given a positive integer \(n\), define \(g(n)\) to be \(\max_{1 \leq k \leq n-1} f(n, k)\).

(a) Given a positive integer \(m\), what is the smallest positive integer \(n_m\) such that \(g(n_m) = m\)?
(b) What is the second smallest?

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