
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
(1) If the inequality $|2x+1|+|x-2| \geqslant a^2-a+\frac{1}{2}$ holds for any real number $x$, find the range of values for $a$.

(2) Given $x, y, z \in \mathbb{R}$, and $x^2+y^2+z^2=14$, find the maximum value of $x+2y+3z$.

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