
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) Let $x$, $y$, $z$ be real numbers, satisfying: $x^2+y^2+z^2=1$ and $x+2y+3z= \sqrt{14}$. Find the value of $x+y+z$;  
(2) Let the solution set of the inequality $|x-2|<a$ (where $a \in \mathbb{N}^*$) be $A$, and $\frac{3}{2} \in A$, $\frac{1}{2} \notin A$. Find the minimum value of the function $f(x)=|x+a|+|x-2|$.

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