
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
Consider the function \(f(x)=|2x-1|\):
1. Solve the inequality \(f(2x)\leqslant f(x+1)\) with respect to \(x\).
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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