
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
What does the formula for the $n$-th term of a $k$-th order recurrence sequence look like if

a) the characteristic equation has simple roots $x_{1}, \ldots, x_{k}$, each different from zero;

b) the characteristic equation has roots different from zero $x_{1}, \ldots, x_{m}$ with multiplicities $\alpha_{1}, \ldots, \alpha_{m}$ respectively.

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