
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 $\{a_k\}_{k=1}^{2023}$ be the sequence of real numbers defined by $a_1=0.123,$ $a_2=(0.1231)^{a_1},$ $a_3=(0.12301)^{a_2},$ $a_4=(0.123011)^{a_3}$, and in general, 
\[a_k=\begin{cases}(0.\underbrace{12301\cdots 0101}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd,}\\(0.\underbrace{12301\cdots 01011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even.}\end{cases}\]
Rearranging the numbers in the sequence  $\{a_k\}_{k=1}^{2023}$ in decreasing order produces a new sequence  $\{b_k\}_{k=1}^{2023}$.  What is the sum of all integers $k$, $1 \le k \le 2023$, such that $a_k = b_k?$
A) 1013112   
B) 1023112   
C) 1033112   
D) 1043112

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