
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 $\{a_k\}_{k=1}^{1000}$ be the sequence of real numbers defined by $a_1= 0.2$, and each subsequent $a_k$ for $k > 1$ given by:
\[a_k = \begin{cases} (0.201)^{a_{k-1}} & \text{if } k \text{ is odd,} \\ (0.202)^{a_{k-1}} & \text{if } k \text{ is even.} \end{cases}\]
Write out the sequence $\{a_k\}_{k=1}^{1000}$ in increasing order to get a new sequence $\{b_k\}_{k=1}^{1000}$. What is the sum of all integers $k$ such that $a_k = b_k$?
A) 740
B) 745
C) 750
D) 755
E) 760

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