
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 real numbers $a>0$, $b>0$, and $a^2+3b^2 = 3$. If $\sqrt{5}a+b \leq m$ always holds, then  
(1) Find the minimum value of m;  
(2) If $2|x-1|+|x| \geq \sqrt{5}a+b$ always holds for $a>0$, $b>0$, determine the range of the real number x.

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