
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 $\{a_k\}_{k=1}^{1005}$ be the sequence of real numbers defined by $a_1=0.3,$ $a_2=(0.301)^{a_1},$ $a_3=(0.3011)^{a_2},$ $a_4=(0.30101)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{30101\cdots 01}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is even,}\\(0.\underbrace{30101\cdots 011}_{k+2\text{ digits}})^{a_{k-1}}\qquad\text{if }k\text{ is odd.}\end{cases}\]
Rearranging the numbers in the sequence $\{a_k\}_{k=1}^{1005}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{1005}$. What is the sum of all integers $k$, $1 \le k \le 1005$, such that $a_k = b_k?$
A) 250000
B) 251000
C) 252000
D) 253009
E) 254000

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