
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
Let the function $f(x)$ have its maximum and minimum values in the interval D defined as $\max\{f(x) \mid x \in D\}$ and $\min\{f(x) \mid x \in D\}$ respectively. The function $f(x)$ is defined as 
$$
f(x)= 
\begin{cases} 
-x+2b & x\in[1,b] \\
b & x\in(b,3] 
\end{cases}
$$
where $1 < b < 3$. The function $g(x) = f(x) + ax$, where $x \in [1, 3]$.
(1) If the function $g(x)$ is monotonically decreasing on $[1, 3]$, find the range of values for $a$.
(2) If $a \in \mathbb{R}$, let $h(a) = \max\{g(x) \mid x \in [1, 3]\} - \min\{g(x) \mid x \in [1, 3]\}$. Define $d(b) = \min\{h(a) \mid a \in \mathbb{R}\}$. Write the expression for $h(a)$ and find $\min\{d(b) \mid b \in (1,3)\}$.
(3) Define $k(a) = \max\{g(f(x)) \mid x \in l\} - \min\{g(f(x)) \mid x \in l\}$, where $l$ is the domain of $g(f(x))$. If $l$ is exactly $[1, 3]$, find the range of values of $b$ and the minimum of $k(a)$ for all $a \in \mathbb{R}$.

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