
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 \( P = \frac{1}{1 \times 2}+\frac{1}{2 \times 3}+\frac{1}{3 \times 4}+\cdots+\frac{1}{99 \times 100} \), find the value of \( P \).

Given that \( 99 Q = P \times\left(1+\frac{99}{100}+\frac{99^{2}}{100^{2}}+\frac{99^{3}}{100^{3}}+\ldots\right) \), find the value of \( Q \).

Given that \( x \) and \( R \) are real numbers and \( \frac{2 x^{2}+2 R x+R}{4 x^{2}+6 x+3} \leq Q \) for all \( x \), find the maximum value of \( R \).

Given that \( S=\log _{144} \sqrt[R]{2}+\log _{144} \sqrt[R]{R} \), find the value of \( S \).

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