
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) = |x - 4a| + |x|, where a ∈ R.
(I) Find the range of values for the real number a if the inequality f(x) ≥ a² is always true for all x ∈ R.
(II) Let m be the maximum value of a from part (I). If the real numbers x, y, and z satisfy the equation 4x + 2y + z = m, find the minimum value of (x + y)² + y² + z².

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