
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 $f(x) = |2x+1| + |x-1|$.
(1) Solve the inequality $f(x) \geq 3$;
(2) Let the minimum value of the function $f(x)$ be $m$. If $a$, $b$, and $c$ are all positive real numbers, and $\frac{1}{2}a + b + 2c = m$, find the minimum value of $a^2 + b^2 + c^2$.

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