
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| - |2x - a|, where a > 1 and a is a real number.

1. Solve the inequality f(x) ≥ (1/2)x when a = 2.
2. If the maximum value of f(x) is M, and positive real numbers b and c satisfy (1/b) + (2/c) = a - M, find the minimum value of (2/(b - 1)) + (1/(c - 2)).

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