
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 \(\left\{\begin{array}{r}w \cdot x \cdot y \cdot z = 4 \\ w - x \cdot y \cdot z = 3\end{array}\right.\) and \(w > 0\). If the solution of \(w\) is \(P\), find the value of \(P\).

Let \([y]\) represent the integral part of the decimal number \(y\). For example, \([3.14]=3\). If \(\left[(\sqrt{2}+1)^{p}\right] = Q\), find the value of \(Q\).

Given that \(x_{0}y_{0} \neq 0\) and \(Q x_{0}^{2} - 22 \sqrt{3} x_{0} y_{0} + 11 y_{0}^{2} = 0\). If \(\frac{6 x_{0}^{2} + y_{0}^{2}}{6 x_{0}^{2} - y_{0}^{2}} = R\), find the value of \(R\).

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