
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 $O$ is the coordinate origin, $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=2 (a > b > 0)$ has left and right foci $F_{1}$ and $F_{2}$, respectively, right vertex $A$, and upper vertex $B$. If $|OB|$, $|OF_{2}|$, and $|AB|$ form a geometric sequence, and the shortest distance from a point on the ellipse $C$ to focus $F_{2}$ is $\sqrt{6} - 2$, find:

1. The standard equation of ellipse $C$;
2. The minimum value of $\frac{|TF_{1}|}{|PQ|}$, where $T$ is any point on the line $x=-3$, line $F_{1}PQ$ passes through $F_{1}$ and is perpendicular to $TF_{1}$, intersecting ellipse $C$ at points $P$ and $Q$.

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