
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 the function $f(x) = |x-1|$, the solution set of the inequality $f(x+5) \leq 3m$ ($m>0$) is $[-7, -1]$.

(1) Find the value of $m$;  
(2) Given $a>0$, $b>0$, and $2a^2+b^2=3m$, find the maximum value of $2a\sqrt{1+b^2}$.

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