
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 equation of the curve $C$ as ${y}^{2}=4x$, a line with slope $l$ is drawn through the origin intersecting the curve $C$ at another point ${P}_{1}$. A line with slope $2$ is drawn through ${P}_{1}$ intersecting the curve $C$ at another point ${P}_{2}$. A line with slope $4$ is drawn through ${P}_{2}$ intersecting the curve $C$ at another point ${P}_{3}$, and so on. In general, a line with slope ${2}^{n}$ is drawn through point ${P}_{n}$ intersecting the curve $C$ at another point ${P}_{n+1}$. Let point ${P}_{n}$ be represented as $({x}_{n},{y}_{n})$ where $n∈\mathbb{N}^*$.

(1) Determine ${y}_{1}$ and find the relationship between ${y}_{n+1}$ and ${y}_{n}$;

(2) Find the general term formula for the sequence ${y}_{2n-1}$, where $n∈\mathbb{N}^*$, and explain which point the sequence of points ${P}_{1}$, ${P}_{3}$, ..., ${P}_{2n+1}$, ... approaches infinitely;

(3) Let ${a}_{n}={y}_{2n+1}-{y}_{2n-1}$. The sum of the first $n$ terms of the sequence ${a}_{n}$ is denoted as ${S}_{n}$. Set ${b}_{n}=\frac{1}{\frac{3}{4}{S}_{n}+1}$. Calculate the sum of all possible products ${b}_{i}·{b}_{j}$ where $1 ≤ i ≤ j ≤ n$.

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