
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
In the Cartesian coordinate plane $(xOy)$, an ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ passes through the point $(1, \frac{4\sqrt{2}}{3})$, and the distance from the right focus $F$ to the left directrix $l$ is $\frac{14}{\sqrt{5}}$.

(1) Find the standard equation of the ellipse;

(2) If point $M(m, n)$ is a point outside the ellipse, and the two tangent lines $(l_{1}, l_{2})$ passing through point $M$ are perpendicular to each other, find the value of $m^{2} + n^{2}$.

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