
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 $P$ is a moving point on the ellipse $\frac{x^2}{8} + \frac{y^2}{4} = 1$ in the first quadrant, through point $P$, two tangents $PA$ and $PB$ are drawn to the circle $x^2 + y^2 = 4$, with the points of tangency being $A$ and $B$ respectively. The line $AB$ intersects the $x$-axis and $y$-axis at points $M$ and $N$ respectively. Find the minimum value of the area of $\triangle OMN$.

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