
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
Given an arithmetic-geometric sequence \( u_{n+1} = a u_{n} + b \):

1. Consider another sequence \(\left(v_{n}\right)\) that satisfies the same recurrence relation as \(\left(u_{n}\right)\) (i.e., \( v_{n+1} = a \cdot v_{n} + b \)). What can be said about the sequence \( \left(w_{n}\right) = \left(u_{n} - v_{n}\right) \)?
2. By finding a simple sequence \(\left(v_{n}\right)\) as mentioned earlier, deduce a formula for \(\left(u_{n}\right)\).

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