
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 is a sequence of points $P_0, P_1, P_2, \cdots, P_{n-1}, P_n$, where the coordinates of point $P_k$ are $(x_k, y_k) (k \in \mathbb{N}, k \leqslant n)$, and $x_k, y_k \in \mathbb{Z}$. Let $\Delta x_k = x_k - x_{k-1}$, $\Delta y_k = y_k - y_{k-1}$, 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 $\{y_k\} (k \in \mathbb{N}^*, k \leqslant n)$ is an increasing sequence, 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 $x_0 + x_1 + x_2 + \cdots + x_{2016}$.

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