
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 a point $M$ moving on the ellipse $\frac{y^{2}}{a^{2}}+ \frac{x^{2}}{b^{2}}=1 (a > b > 0)$, two tangent lines are drawn from $M$ to the circle $x^{2}+y^{2}= \frac{b^{2}}{2}$. Let $P$ and $Q$ be the points of tangency, and $E$ and $F$ be the intersection points of line $PQ$ with the $x$-axis and $y$-axis, respectively. Find the minimum area of $\triangle EOF$.

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