
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
Darya Dmitrievna is preparing a test on number theory. She promised each student to give as many problems as the number of terms they create in the numerical example

$$
a_{1} + a_{2} + \ldots + a_{n} = 2021
$$

where all numbers \( a_{i} \) are natural, greater than 10, and are palindromes (they do not change if their digits are written in reverse order). If a student does not find any such example, they will receive 2021 problems on the test. What is the smallest number of problems a student can receive? (20 points)

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