
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 $x^2+y^2+z^2=1$.

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

(Ⅱ) If the inequality $|a-3| \geq x+2y+2z$ holds for all positive numbers $x$, $y$, $z$, find the range of the real number $a$.

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