
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 the domain of the function $y=f(x)$ is $\mathbb{R}$, and for any $x$ in the domain, there exists a real number $a$ such that $f=f(x+a)=f(-x)$ holds. We say that such a function has the "$P(a)$ property".
(1) Determine whether the function $y=\sin x$ has the "$P(a)$ property". If it has the "$P(a)$ property", write out all the values of $a$; if not, explain the reason.
(2) It is known that $y=f(x)$ has the "$P(0)$ property", and when $x\leqslant 0$, $f(x)=(x+t)^{2}$, $t\in \mathbb{R}$. Find the maximum value of $y=f(x)$ on $[0,1]$.
(3) Suppose that the function $y=g(x)$ has the "$P(±1)$ property", and when $- \frac {1}{2}\leqslant x\leqslant \frac {1}{2}$, $g(x)=|x|$. Find: the analytic expression of function $g(x)$ when $x∈\mathbb{R}$, and the value of $m$ if the number of intersection points of $y=g(x)$ and $y=mx (m∈\mathbb{R})$ is $1001$.

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