
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 $a, b, c \in \mathbb{R}^+$, and $a+ \sqrt{2}b+ \sqrt{3}c=2 \sqrt{3}$, let the minimum value of $a^2+b^2+c^2$ be $m$.  
(Ⅰ) Find the real number $m$;  
(Ⅱ) If the solution set of the inequality $|x-3| \geq m$ and $x^2+px+q \geq 0$ are the same, find the value of $p$.

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