
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 \(\{a_k\}_{k=1}^{1011}\) of real numbers defined by
\( a_1 = 0.202, a_2 = (0.2021)^{a_1}, a_3 = (0.20201)^{a_2}, a_4 = (0.202011)^{a_3} \),
and in general,
\[
a_k=\begin{cases}
(0.\underbrace{20201\cdots 0201}_{k+2\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is odd, } \\
(0.\underbrace{20201\cdots 02011}_{k+2\text{ digits}})^{a_{k-1}} & \text{if } k \text{ is even.}
\end{cases}
\]
calculate the sum of all integers \( k \), \( 1 \leq k \leq 1011 \), such that \( a_k \) is the smallest term among the successive three terms \( a_{k-1}, a_k, a_{k+1} \).

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