
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 an ellipse \( T: \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 \) with \( a > b > 0 \) and a hyperbola \( S: \frac{x^{2}}{m^{2}} - \frac{y^{2}}{n^{2}} = 1 \) with \( m > 0 \), \( n > 0 \), both having the same focus \( F(2,0) \). Let the hyperbola \( S \) pass through the asymptotic line \( l \) in the first quadrant. If the focus \( F \) and the upper vertex \( B \) of the ellipse \( T \) have symmetrical points with respect to line \( l \), and these symmetrical points lie on the hyperbola \( S \), find the equations of the ellipse \( T \) and the hyperbola \( S \).

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