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

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