
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

We have 3 locations labeled as 1, 2, and 3 where disks can be placed. Initially, there are \( n \) disks on position 1, with each disk having a smaller radius than the one below it. The goal is to move all these disks from position 1 to position 2 while respecting the following 2 rules:

1. At each step, exactly one disk can be moved from one position to another.
2. A disk cannot be placed on top of another disk with a smaller radius.

Show by induction that all the disks can be moved to position 2, and find a formula, also by induction, for the minimum number of steps required.

Additionally, consider an arithmetic-geometric sequence defined as \(\left(u_{n}\right)_{n \in \mathbb{N}}\) by the first term (denoted as \(u_{1}\)) and the recurrence relation \(u_{n+1}=a \cdot u_{n}+b\), where \(a \neq 1\) and \(b\) are fixed.

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