
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 the average weight of the quarters in $P(Q)$ is $30$ grams, the average weight in $R(Q)$ is $40$ grams, and when combined, piles $P(Q)$ and $R(Q)$ have an average weight of $34$ grams, and when piles $P(Q)$ and $S(Q)$ are combined, the average weight is $35$ grams, determine the maximum possible integer value for the average weight in grams of the quarters in the combined piles $R(Q)$ and $S(Q)$.

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