
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 point $P(-3,1)$ on the left directrix of the ellipse $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a > b > 0)$, the line $l$ passing through point $P$ is $5x + 2y = 13$. After reflecting on the line $y=-2$, the line passes through the left focus of the ellipse. Find the eccentricity of this ellipse.

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