
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
Using a balance to measure the weight of items with integer masses between $1$ and $100$, if weights can be placed only on one side of the balance, at least how many weights are needed? If weights can be placed on both sides of the balance, at least how many weights are needed?

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