
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) = |3x + 1| - |2x + 2|:
1. Find the solution set for the inequality f(x) ≥ 0.
2. If f(x) - |x + 1| ≤ |a + 1| holds true for any x ∈ R, find the range of real number values for a.

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