
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 with an eccentricity of \( e = \frac{2}{5} \sqrt{5} \), passing through the point \( (1,0) \) and tangential to the line \( l: 2x - y + 3 = 0 \) at the point \( P\left( -\frac{2}{3}, \frac{5}{3} \right) \), with its major axis parallel to the \( y \)-axis, find the equation of this ellipse.

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