
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
In the Cartesian coordinate system, there are points $P_0$, $P_1$, $P_2$, $P_3$, ..., $P_{n-1}$, $P_n$. Let the coordinates of point $P_k$ be $(x_k,y_k)$ $(k\in\mathbb{N},k\leqslant n)$, where $x_k$, $y_k\in\mathbb{Z}$. Denote $\Delta x_k=x_k-x_{k-1}$, $\Delta y_k=y_k-y_{k-1}$, and it satisfies $|\Delta x_k|\cdot|\Delta y_k|=2$ $(k\in\mathbb{N}^*,k\leqslant n)$;  
(1) Given point $P_0(0,1)$, and point $P_1$ satisfies $\Delta y_1 > \Delta x_1 > 0$, find the coordinates of $P_1$;  
(2) Given point $P_0(0,1)$, $\Delta x_k=1$ $(k\in\mathbb{N}^*,k\leqslant n)$, and the sequence $\{y_k\}$ $(k\in\mathbb{N},k\leqslant n)$ is increasing, point $P_n$ is on the line $l$: $y=3x-8$, find $n$;  
(3) If the coordinates of point $P_0$ are $(0,0)$, and $y_{2016}=100$, find the maximum value of $x_0+x_1+x_2+…+x_{2016}$.

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