
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 E: \dfrac{x^2}{a^2}+ \dfrac{y^2}{b^2}=1(a > b > 0), and the point \left( 1, \dfrac{ \sqrt{2}}{2} \right) is on the ellipse. Also, the distance from the point (-1,0) to the line PF_2 is \dfrac{4 \sqrt{5}}{5}, where point P(-1,-4). Determine the equation of the ellipse E.

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