
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 that \\(F_1\\) and \\(F_2\\) are the left and right foci of the ellipse \\(\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)\\), and \\(B_1\\), \\(B_2\\) are the endpoints of the short axis of the ellipse, the quadrilateral \\(F_1B_1F_2B_2\\) is a square and its area is \\(50\\).

\\((1)\\) Find the equation of the ellipse;

\\((2)\\) A line \\(l\\) passing through the focus \\(F_1\\) with an inclination angle of \\(30^\circ\\) intersects the ellipse at points \\(M\\) and \\(N\\). Find the radius of the inscribed circle of triangle \\(\Delta F_2MN\\).

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