
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 the parabola $y^2=4x$ with focus as the right focus $F$ of the ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a > b > 0)$. Point $B$ is the intersection of this parabola and ellipse $C$ in the first quadrant, and $|BF|=\frac{5}{3}$.

$(1)$ Find the equation of ellipse $C$;

$(2)$ Draw two perpendicular lines $l_1$ and $l_2$ through point $F$. Line $l_1$ intersects ellipse $C$ at points $P$ and $Q$, and line $l_2$ intersects the line $x=4$ at point $T$. Find the range of $\frac{|TF|}{|PQ|}$.

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