
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

João has a single chain with $n$ links, each weighing $1 \mathrm{~g}$. Every time a link in any piece of the chain is broken, we get 3 pieces. For example, if a piece has 9 links and we break the fourth one, we will have pieces with the following number of links: 3, 1, and 5. A broken link still weighs $1 \mathrm{~g}$.

a) If $n=8$, show an example of how to break only 2 links to obtain pieces that can be combined to measure any weight from $1 \mathrm{~g}$ to $8 \mathrm{~g}$.

b) If $n=16$, show an example of how to break only 2 links to obtain pieces that can be combined to measure any weight from $1 \mathrm{~g}$ to $16 \mathrm{~g}$.

c) If $n=63$, show an example of how to break only 3 links to obtain pieces that can be combined to measure any weight from $1 \mathrm{~g}$ to $63 \mathrm{~g}$.

d) If $n=k \cdot 2^{k}-1$, show an example of how to break only $k-1$ links to obtain pieces that can be combined to measure any weight from $1 \mathrm{~g}$ to $\mathrm{ng}$.

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