
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 \(x\), \(y\), \(z\) satisfying \(x^{2}+y^{2}+z^{2}=6\).

(Ⅰ) Find the maximum value of \(x+2y+z\);

(Ⅱ) If the inequality \(|a+1|-2a \geqslant x+2y+z\) always holds for \(x\), \(y\), \(z\) under the given condition, find the range of the real number \(a\).

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