
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 the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with an eccentricity of $\frac{\sqrt{2}}{2}$, the long axis $AB$ is divided into $2016$ equal parts. From left to right, these points are sequentially named $M_{1}$, $M_{2}$, $…$, $M_{2015}$. A line with a slope of $k(k\neq 0)$ is drawn through point $M_{1}$, intersecting the ellipse $C$ at points $P_{1}$ and $P_{2}$, with $P_{1}$ located above the $x$-axis. Similarly, a line with a slope of $k(k\neq 0)$ is drawn through point $M_{2}$, intersecting the ellipse $C$ at points $P_{3}$ and $P_{4}$, with $P_{3}$ located above the $x$-axis. This process is repeated until a line with a slope of $k(k\neq 0)$ is drawn through point $M_{2015}$, intersecting the ellipse $C$ at points $P_{4029}$ and $P_{4030}$, with $P_{4029}$ located above the $x$-axis. Find the product of the slopes of the $4030$ lines $AP_{1}$, $AP_{2}$, $…$, $AP_{4030}$.

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