
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  $P$  be the quadratic function such that  $P(0) = 7$ ,  $P(1) = 10$ , and  $P(2) = 25$ .  If  $a$ ,  $b$ , and  $c$  are integers such that every positive number  $x$  less than 1 satisfies
\[
  \sum_{n = 0}^\infty P(n) x^n = \frac{ax^2 + bx + c}{{(1 - x)}^3},
\]
compute the ordered triple  $(a, b, c)$ .

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