
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
$$
f(x)=a x+b, \quad g(x)=\left\{\begin{array}{ll}
f(x), & 0 \leq x \leq a, \\
f(f(x)), & x>a .
\end{array}\right.
$$

If for any positive number $a$, the function $g(x)$ is a strictly increasing function on $[0,+\infty)$, what is the minimum value of the constant $b$?

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