
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: $m=\frac{{|{a+b}|}}{c}+\frac{{2|{b+c}|}}{a}+\frac{{3|{c+a}|}}{b}$, and $abc > 0$, $a+b+c=0$. Determine $x+y$ where $x$ is the number of different values that $m$ can take and $y$ is the largest of these values.

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