
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) There are 128 coins of two different weights, with an equal number of coins of each weight. How can you guarantee finding two coins of different weights using no more than seven weighings on a balance scale without weights?

b) There are eight coins of two different weights, with an equal number of coins of each weight. How can you guarantee finding two coins of different weights using two weighings on a balance scale without weights?

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