
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
a) Consider a prime $p$ that divides $10^{n} + 1$ for some positive integer $n$. For example, $p = 7$ divides $10^{3} + 1$. By analyzing the main period of the decimal representation of $\frac{1}{p}$, verify that the number of times the digit $i$ appears is equal to the number of times the digit $9 - i$ appears for each $i \in \{0, 1, 2, \ldots, 9\}$.

b) Consider a prime number $p$ that does not divide 10 and suppose that the period of the decimal representation of $\frac{1}{p}$ is $2k$. Is it always possible to break the period into two numbers that sum up to $10^k - 1$? For example, the period of $\frac{1}{7}$ has a length of $6 = 2k$, as it is equal to 142857. See that $142 + 857 = 999 = 10^3 - 1 = 10^k - 1$.

c) Given

$$
x = \frac{1}{1998} + \frac{1}{19998} + \frac{1}{199998} + \ldots
$$

when we write $2x$ as a decimal number, what is the $59^{\text{th}}$ digit after the decimal point?

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