
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 \( \{a_k\}_{k=1}^{101}\) be the sequence of real numbers defined by \( a_1 = 0.202 \), \( a_2 = (0.2021)^{a_1} \), and in general,
\[
a_k = \begin{cases} 
(0.\underbrace{2020\cdots 20}_{k+3 \text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd}, \\
(0.\underbrace{2020\cdots 202}_{k+3 \text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even}.
\end{cases}
\]
Rearranging the numbers in the sequence  \( \{a_k\}_{k=1}^{101} \) in decreasing order produces a new sequence  \( \{b_k\}_{k=1}^{101} \). What is the sum of all integers \( k \), \( 1 \le k \le 101 \), such that \( a_k = b_k \)?
A) 2451
B) 2550
C) 2651
D) 2750
E) 2851

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