
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 the ellipse $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$, let $P$ be a point on the ellipse such that $\angle F_{1}PF_{2}=60^{\circ}$, and $S_{\triangle PF_{1}F_{2}}= \frac{\sqrt{3}}{3}$.
(I) Find $b$;
(II) If $a=2$, $A(0,b)$, does there exist an isosceles right triangle inscribed within the ellipse with $A$ as its right angle vertex? If so, find the number of such triangles. If not, explain the reason.

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