
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
Suppose $A$ and $B$ are two moving points on a parabola, and the tangents to the parabola at $A$ and $B$ are perpendicular to each other. It is known that the locus of the centroid of the triangle formed by $A$, $B$, and the vertex of the parabola is also a parabola, denoted as $L\_1$. Repeating the above process on $L\_1$ yields another parabola $L\_2$, and so on. The sequence of parabolas obtained in this way is denoted as $L\_1$, $L\_2$, $L\_3$, $L\_4$, ..., $L\_n$. If the equation of the parabola is $y^{2}=6x$, it has been calculated that $L\_1$: $y^{2}=2(x-1)$, $L\_2$: $y^{2}=\\frac{2}{3} (x-1-\\frac{1}{3} )=\\frac{2}{3} (x-\\frac{4}{3} )$, $L\_3$: $y^{2}=\\frac{2}{9} (x-1-\\frac{1}{3} -\\frac{1}{9} )=\\frac{2}{9} (x-\\frac{13}{9} )$, $L\_4$: $y^{2}=\\frac{2}{27} (x-1-\\frac{1}{3} -\\frac{1}{9} -\\frac{1}{27} )=\\frac{2}{27} (x-\\frac{40}{27} )$, ..., $L\_n$: $y^{2}=\\frac{2}{{S}\_{n}} (x-\\frac{{T}\_{n}}{{S}\_{n}} )$. Find $2T\_n-3S\_n=$*______*.

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