
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 $\{a_k\}_{k=1}^{11}$ of real numbers defined by $a_1=0.5$, $a_2=(0.51)^{a_1}$, $a_3=(0.501)^{a_2}$, $a_4=(0.511)^{a_3}$, and in general,

$a_k=\begin{cases}
(0.\underbrace{501\cdots 01}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd,} \\
(0.\underbrace{501\cdots 011}_{k+1\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even.}
\end{cases}$

Rearrange the numbers in the sequence $\{a_k\}_{k=1}^{11}$ in decreasing order to produce a new sequence $\{b_k\}_{k=1}^{11}$. Find the sum of all integers $k$, $1\le k \le 11$, such that $a_k = b_k$.

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