
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 the ellipse $\Gamma: \frac{x^{2}}{4} + y^{2} = 1$ be in the rectangular coordinate system $xOy$. Point $L$ is in the third quadrant and $OL = 3\sqrt{2}$. The angle between the negative x-axis and $OL$ is $\frac{\pi}{4}$. Two tangents $l_{1}$ and $l_{2}$ to $\Gamma$ pass through point $L$ and intersect the y-axis at points $P$ and $Q$, respectively. Find the value of $|OP| \cdot |OQ|$.

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