
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]


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Given the hyperbola $E: \frac{x^2}{a^2} - \frac{y^2}{a^2-4} = 1$ ($a>2$),
1. If the eccentricity of $E$ is $\frac{\sqrt{14}}{3}$, find the equation of $E$.
2. Let the left and right foci of $E$ be $F_1$ and $F_2$, respectively, and point $P$ be a point on the hyperbola. The line through $F_2$ and $P$ intersects the y-axis at point $Q$, and $F_1P \perp F_1Q$. Does point $P$ lie on a certain fixed line when $a$ varies? If such a fixed line exists, find the equation of the line; if not, explain why.

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