
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 $\{a_k\}_{k=1}^{11}$ be the sequence of real numbers defined by $a_1=0.3,$ $a_2=(0.31)^{a_1},$ $a_3=(0.301)^{a_2},$ $a_4=(0.3011)^{a_3}$, and in general,
\[a_k=\begin{cases}(0.\underbrace{301\cdots 01}_{k+1\text{ digits}})^{a_{k-1}}\quad\text{if }k\text{ is odd,}\\(0.\underbrace{301\cdots 011}_{k+1\text{ digits}})^{a_{k-1}}\quad\text{if }k\text{ is even.}\end{cases}\]
Rearranging the numbers in sequence $\{a_k\}_{k=1}^{11}$ in decreasing order produces a new sequence $\{b_k\}_{k=1}^{11}$. What is the sum of all integers $k$, $1\le k \le 11$, such that $a_k = b_k?$
A) 20  
B) 31  
C) 41  
D) 51  
E) 61

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