
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{4}+ \frac{y^2}{3}=1$, respectively. A point $P$ with coordinates $(x, y)$ lies on the line $y-x-3=0$ (where $x \neq -3$ and $x \neq \pm \sqrt{3}$. The slopes of lines $PF_1$ and $PF_2$ are $k_1$ and $k_2$, respectively. Find the value of $\frac{1}{k_2} - \frac{2}{k_1}$.

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