
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
Let the function $f(x) = |2x-1|$.  
(1) Solve the inequality $f(2x) \leq f(x+1)$.  
(2) If real numbers $a$ and $b$ satisfy $a+b=2$, find the minimum value of $f(a^2) + f(b^2)$.

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