
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 $x$, $y$, $z$ that satisfy $x^{2}+y^{2}+z^{2}=6$:
$(1)$ Find the maximum value of $x+2y+z$;
$(2)$ If the inequality $|a+1|-2a \geqslant x+2y+z$ is always true for any $x$, $y$, $z$ that meet the condition, find the range of the real number $a$.

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