
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 $C : \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with foci $F_{1}\left(-1,0\right), F_{2}\left(1,0\right)$, and the ellipse passes through point $A\left( \frac{4}{3}, \frac{1}{3}\right)$.

(1) Find the eccentricity of the ellipse $C$.

(2) The line $l$ passing through point $F_{2}$ intersects the ellipse $C$ at points $P$ and $Q$, with $\overrightarrow{{F}\_{1}P} \perp \overrightarrow{{F}\_{1}Q}$. Find the equation of the line $l$.

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