
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 that $a$ and $b$ are additive inverses, $b$ and $c$ are multiplicative inverses, and $m^3 = m$.
1. Determine the value of $\frac{2a+2b}{m+2} + ac$.
2. If $a > 1$ and $m < 0$, let $S = |2a - 3b| - 2|b - m| - |b + \frac{1}{2}|$. Find the value of $4(2a - S) + 2(2a - S) - (2a - S)$.
3. If $m \neq 0$, discuss whether $|x + m| - |x - m|$ has a maximum value for rational $x$. If it does, find the maximum value; if not, give a reason.

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