
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 mean weight of the rocks in $A$ is $35$ pounds, the mean weight of the rocks in $B$ is $45$ pounds, the mean weight of the rocks in the combined piles $A$ and $B$ is $38$ pounds, and the mean weight of the rocks in the combined piles $B$ and $C$ is $42$ pounds, calculate the greatest possible integer value for the mean in pounds of the rocks in the combined piles $A$ and $C$.

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