
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 positive real numbers $a$ and $b$ satisfying $ab=3$, find the minimum value of $a+3b$ to be filled in the blank. If the inequality $m^{2}-(a+3b)m+5\leqslant 0$ always holds for the given conditions of $a$ and $b$, determine the range of real number $m$ to be filled in the blank.

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