
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
Let the function $f(x) = |x-1| - 2|x+1|$ have a maximum value of $m$.  
(I) Find the value of $m$;  
(II) If $a, b, c \in (0, +\infty)$, and $a^2 + 3b^2 + 2c^2 = m$, find the maximum value of $ab + 2bc$.

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