
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 real numbers $a (a \neq 0)$ and $b$, the inequality $|a+b|+|a-b| \geqslant M \cdot |a|$ always holds. Let $m$ be the maximum value of the real number $M$.
(1) Find the value of $m$;
(2) Solve the inequality $|x-1|+|x-2| \leqslant m$.

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