
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 sequence \(\left\{a_{n}\right\}\) consisting of 1007 terms with the general term formula:
\[ a_{n} = 10^{n} + 10^{2016-n} \quad (1 \leq n \leq 1007), \]
find the solutions to the following congruences in the sequence \(\left\{a_{n}\right\}\) (given that 2017 is a prime number and 10 is a primitive root modulo 2017):

\[
\begin{array}{l}
(1) \; x \equiv 1 \pmod{2017}; \\
(2) \; x^{2} \equiv 2 \pmod{2017}; \\
(3) \; x^{3} - 3x \equiv 1 \pmod{2017}; \\
(4) \; x^{3} + x^{2} - 2x \equiv 1 \pmod{2017}.
\end{array}
\]

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