
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
A balance has unequal arms and pans of unequal weight. It is used to weigh two objects of unequal weight. The first object balances against a weight \( A \) when placed in the left pan and against a weight \( a \) when placed in the right pan. The corresponding weights for the second object are \( B \) and \( b \). A third object balances against a weight \( C \) when placed in the left pan. What is its true weight?

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