
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
Rationalize the denominator of $\frac{7}{3+\sqrt{8}}$. Simplify the expression to the form $\frac{P\sqrt{Q}+R}{S}$, where $P$, $Q$, $R$, and $S$ are integers, $S$ is positive, and $Q$ is not divisible by the square of any prime. After confirming that the greatest common divisor of $P$, $R$, and $S$ is 1, calculate $P+Q+R+S$.

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