
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
For any two unequal real numbers \(a\) and \(b\), the inequality \(M|a-b| \leq |2a+b|+|a+2b|\) always holds. Let the maximum value of the real number \(M\) be \(m\).
1. Find the value of \(m\);
2. Solve the inequality \(|x-1| < m(2x+1)\).

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