
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 function $f(x)$ is an odd function defined on $[-1,1]$ and $f(1)=1$. If $x,y\in[-1,1]$ and $x+y\neq0$, there is $(x+y)\cdot[f(x)+f(y)]>0$.

(1) Determine the monotonicity of $f(x)$ and provide a proof.
(2) Solve the inequality $f(x+\frac{1}{2}) < f(1-2x)$.
(3) If $f(x)\leqslant m^{2}-2am+1$ holds for all $x\in[-1,1]$ and $a\in[-1,1]$, find the range of values for the real number $m$.

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