
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
Let  $x_0, a, b$  be reals given such that  $b > 0$  and  $x_0 \ne 0$ . For every nonnegative integer  $n$  a real value  $x_{n+1}$  is chosen that satisfies   $$ x^2_{n+1}= ax_nx_{n+1} + bx^2_n . $$  a) Find how many different values  $x_n$  can take.
b) Find the sum of all possible values of  $x_n$  with repetitions as a function of  $n, x_0, a, b$ .

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