
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
If the function $y=f(x)$ has a domain of $\mathbb{R}$, and for any $x$ within this domain there exists a real number $a$ such that $f(x)=f(x+a)=f(-x)$, then this function is said to have the property "P(a)".  
(1) Determine whether the function $y=\sin(x)$ has the property "P(a)". If so, list all values of $a$. If not, provide a reason.  
(2) It is known that $y=f(x)$ has property "P(0)", and for $x\leq 0$, $f(x) = (x+t)^2$ where $t\in\mathbb{R}$. Find the maximum value of $y=f(x)$ on the interval $[0,1]$.  
(3) Suppose the function $y=g(x)$ has the property "P(±1)", and when $-\frac{1}{2} \leq x \leq \frac{1}{2}$, $g(x) = |x|$. Find the analytical expression for $g(x)$ when $x\in\mathbb{R}$. If $y=g(x)$ and $y=mx$ ($m\in\mathbb{R}$) intersect at 1001 points, find the value of $m$.

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