
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
Discuss the inequalities given the functions \(f(x)=|2x+1|\) and \(g(x)=|ax|\).

(1) When \(a =1\), solve the inequality \(f(x) \geqslant g(x)+1\);

(2) When \(a =2\), if for all \(x \in \mathbb{R}\), \(f(x)+g(x) \geqslant b\) always holds, find the range of values for the real number \(b\).

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