
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 parabola C₁ with its focus on the y-axis and passing through the point (2,1), and an ellipse C₂ with foci F₁ and F₂, where F₂ coincides with the focus of C₁. The line passing through F₁ and perpendicular to the major axis of C₂ intersects C₂ at points A and B with |AB|=3. The curve C₃ is a circle with the coordinate origin O as its center and |OF₂| as its radius.
(1) Find the standard equations of C₂ and C₃;
(2) If a moving line l is tangent to C₃ and intersects C₂ at points M and N, find the range of the area S of ∆OMN.

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