
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
Two jars contain different volumes of alcohol solutions. The first jar, with a volume of $V_1$, has a ratio of the volume of alcohol to the volume of water of $r : 1$, and the second jar, with a volume of $V_2$, has a ratio of $s : 1$. Determine the ratio of the volume of alcohol to the volume of water in the mixture when the contents of the two jars are mixed together.
A) $\frac{r+s}{2}$
B) $\frac{rs}{r+s}$
C) $\frac{r+s+2rs}{r+s+2}$
D) $\frac{2(rs + r + s)}{3(r+s)}$
E) $\frac{\frac{r}{r+1}V_1 + \frac{s}{s+1}V_2}{\frac{1}{r+1}V_1 + \frac{1}{s+1}V_2}$

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