
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 $f(x)$ is a function defined on the interval \([-1,1]\) and $f(1)=1$. If $m,n\in[-1,1]$ and $m-n\neq 0$, then $$\frac{f(m)-f(n)}{m-n}<0.$$  
(1) Determine the monotonicity of the function and explain the reason;  
(2) Solve the inequality: $f\left(x+\frac{1}{2}\right)<f(1-x)$;  
(3) If the inequality $f(x)\geq t^2-2at+1$ holds for all $x\in[-1,1]$ and all $t\in[1,2]$, find the range of the real number $a$.

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