
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 alcohol solutions in different ratios: jar A has a ratio of $r:1$ of alcohol to water and jar B has a ratio of $s:1$. The total volume of jar A is $2V_1$, and the volume of jar B is $3V_1$. If an additional volume of $V_1$ of alcohol is added to jar A before the contents of both jars are mixed together, what is the ratio of the total volume of alcohol to the total volume of water in the mixture?
A) $\frac{\frac{2r + 2s}{r + s}}{2}$
B) $\frac{\frac{2r}{r+1} + 1 + \frac{3s}{s+1}}{\frac{2}{r+1} + \frac{3}{s+1}}$
C) $\frac{2(r+s)}{r+s+2}$
D) $\frac{r+s}{2}$
E) $\frac{rs}{r+s}$

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