
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
Let $p^2 = \frac{25}{50}$ and $q^2 = \frac{(3+\sqrt{7})^2}{14}$, where $p$ is a negative real number and $q$ is a positive real number. If $(p+q)^3$ can be expressed in the form $\frac{x\sqrt{y}}{z}$ where $x$, $y$, and $z$ are integers, find the value of the sum $x+y+z$.

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