
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 \( f(x) = \sqrt[3]{1+x} - \lambda x \), where \( \lambda > 0 \).
1. Find the range of values for \( \lambda \) such that \( f(x) \) is a monotonic function on the interval \([0, +\infty)\).
2. Can this monotonic property be extended to the entire domain \((-\infty, +\infty)\)?
3. Solve the inequality \( 2x - \sqrt[3]{1+x} < 12 \).

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