
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) = |2x - 1| - a,

1. Solve the inequality f(x) > x + 1 when a = 1.
2. Find the range of values for the real number a such that there exists a real number x that satisfies f(x) < 0.5 * f(x + 1).

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