
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 $f(x) = |x-3| - 2|x+1|$, its maximum value is $m$.
(1) Find the value of $m$ and the solution set of the inequality $f(x) < 1$;
(2) If $a, b \in (0, +\infty)$, and $a^2 + 2b^2 + c^2 = m$, find the maximum value of $ab + bc$.

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