
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 $a \gt 0$, $b \gt 0$.  
$(Ⅰ)$ If the inequality $\frac{3}{a}+\frac{1}{b}\geqslant \frac{m}{a+3b}$ always holds, find the maximum value of $m$;  
$(Ⅱ)$ If $a+2b+2ab=8$, find the minimum value of $a+2b$.

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