
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 $y = x^a$ has the following properties: if the constant $a > 0$, then the function is decreasing on the interval $(0, c)$ and increasing on the interval $(c, +\infty)$.  
(1) If the function $y = x^b$ is decreasing on the interval $(0, 4)$ and increasing on the interval $(4, +\infty)$, find the value of the real constant $b$;  
(2) Let the constant $c \in (1, 4)$, find the maximum and minimum values of the function $f(x) = x^c$ for $1 \leq x \leq 2$.

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