
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 \( f(x) = x^2 + bx + c \). Let \( C \) be the curve \( y = f(x) \) and let \( P_i \) be the point \( (i, f(i)) \) on \( C \). Let \( A_i \) be the point of intersection of the tangents at \( P_i \) and \( P_{i+1} \). Find the polynomial of smallest degree passing through \( A_1, A_2, \ldots, A_9 \).

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