
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 standard equation of the ellipse \( C \) is \( \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) where \( a > b > 1 \). Point \( P \) is a moving point on the unit circle, point \( A \) and point \( F \) are the right vertex and the right focus of the ellipse \( C \) respectively. Given that when \( \angle POA = \frac{\pi}{3} \), \( |PF| = 1 \) and \( |PA| = \sqrt{3} \):
(1) Find the standard equation of the ellipse \( C \).
(2) Through point \( P \), draw a line \( l \) that intersects the ellipse \( C \) at points \( M \) and \( N \), such that \( P \) is the midpoint of the line segment \( MN \). Then, draw the tangents to the ellipse at points \( M \) and \( N \) which intersect at point \( Q \). Find the range of \( |OQ| \) as point \( P \) moves on the unit circle.

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