
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 $f(x)= \frac{ax}{x+b}$ that satisfies: $f(1)=1$, $f(-2)=4$.
(1) Find the values of $a$ and $b$, and investigate whether there exists a constant $c$ such that for any $x$ in the domain of function $f(x)$, $f(x)+f(c-x)=4$ holds;
(2) When $x \in [1,2]$, the inequality $f(x) \leqslant \frac{2m}{(x+1)|x-m|}$ always holds. Find the range of values for the real number $m$.

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