
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
In a math interest class, the teacher gave a problem for everyone to discuss: "Given real numbers $a$, $b$, $c$ not all equal to zero satisfying $a+b+c=0$, find the maximum value of $\frac{|a+2b+3c|}{\sqrt{{a}^{2}+{b}^{2}+{c}^{2}}}."$ Jia quickly offered his opinion: Isn't this just the Cauchy inequality? We can directly solve it; Yi: I am not very clear about the Cauchy inequality, but I think we can solve the problem by constructing the dot product of vectors; Bing: I am willing to try elimination, to see if it will be easier with fewer variables; Ding: This is similar to the distance formula in analytic geometry, can we try to generalize it to space. Smart you can try to use their methods, or design your own approach to find the correct maximum value as ______.

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