
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 two-dimensional Cartesian coordinate system, there is a sequence of points ${P_0, P_1, P_2, ..., P_{n-1}, P_n}$, where the coordinates of point ${P_k}$ are ${(x_k, y_k) (k \in \mathbb{N}, k \leq n)}$, with ${x_k, y_k \in \mathbb{Z}}$. Let ${\Delta x_k = x_k - x_{k-1}}$, ${\Delta y_k = y_k - y_{k-1}}$, and ${|\Delta x_k| \cdot |\Delta y_k| = 2 (k \in \mathbb{N}^*, k \leq n)}$.

(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 ${P_0(0, 1)}$, ${\Delta x_k = 1 (k \in \mathbb{N}^*, k \leq n)}$, and ${\{y_k\} (k \in \mathbb{N}, k \leq n)}$ is an increasing sequence, and 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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