
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 the polynomial 
$$
\begin{aligned}
f(x)= & a_{2020} x^{2020}+a_{2019} x^{2019}+\cdots+ \\
& a_{2018} x^{2018}+a_{1} x+a_{0},
\end{aligned}
$$

where $a_{i} \in \mathbf{Z}$ for $i=0,1, \cdots, 2020$, determine the number of ordered pairs $(n, p)$ (where $p$ is a prime) such that $p^{2}<n<2020$ and for any $i=0, 1, \cdots, n$, it holds that $\mathrm{C}_{n}^{i}+f(i) \equiv 0 \pmod{p}$.

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