
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 the function \( f(x) = ax + b \) and the piecewise function \( g(x) = \begin{cases} 
f(x), & 0 \leq x \leq a \\
f(f(x)), & x > a 
\end{cases} \), if for any positive number \( a \), the function \( g(x) \) is a strictly increasing function on the interval \( (0, +\infty) \), determine the minimum value of the constant \( b \).

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