
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 a left pan, a right pan, and a pointer that moves along a graduated ruler. Like many other grocer balances, this one works as follows: An object of weight  $ L$  is placed in the left pan and another of weight  $ R$  in the right pan, the pointer stops at the number  $ R \minus{} L$  on the graduated ruler. There are  $ n, (n \geq 2)$  bags of coins, each containing  $ \frac{n(n\minus{}1)}{2} \plus{} 1$  coins. All coins look the same (shape, color, and so on).   $ n\minus{}1$  bags contain real coins, all with the same weight. The other bag (we don’t know which one it is) contains false coins.  All false coins have the same weight, and this weight is different from the weight of the real coins. A legal weighing consists of placing a certain number of coins in one of the pans, putting a certain number of coins in the other pan, and reading the number given by the pointer in the graduated ruler. With just two legal weighings it is possible to identify the bag containing false coins. Find a way to do this and explain it.

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