
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 $f(x) = 4x - x^4$,  
find the intervals where $f(x)$ is monotonic;  
if the equation $f(x) = a$ (where $a$ is a real number) has two distinct roots $x_1$ and $x_2$, and $1 < x_2$, find the maximum value of $x_2 - 1$.

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