
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
Given positive numbers $a$, $b$, $c$ satisfying $a^2+b^2+c^2=6$.

(Ⅰ) Find the maximum value $M$ of $a+2b+c$;  
(Ⅱ) Under the condition of (Ⅰ), if the inequality $|x+1|+|x+m| \geq M$ always holds, find the range of the real number $m$.

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