
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 that the function $f(x)= \sqrt{2|x-3|-|x|-m}$ has a domain of $\mathbb{R}$;  
(I) Find the range of values for the real number $m$;  
(II) Let the real number $t$ be the maximum value of $m$, and let real numbers $a$, $b$, $c$ satisfy $a^2+b^2+c^2=t^2$. Determine the minimum value of $\frac{1}{a^2+1}+\frac{1}{b^2+2}+\frac{1}{c^2+3}$.

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