
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 \(\{a_k\}_{k=1}^{100}\) defined by \(a_1 = 1\) and \(a_k = \begin{cases} a_{k-1} + 2 & \text{if } k \text{ is odd,}\\ (a_{k-1} + 2)^2 & \text{if } k \text{ is even,} \end{cases}\), rearrange the numbers in the sequence \(\{a_k\}_{k=1}^{100}\) in increasing order to form a new sequence \(\{b_k\}_{k=1}^{100}\) and find the sum of all integers \(k, 1 \leq k \leq 100\), such that \(a_k = b_k\).

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