
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 the sequence of positive integers \(\{a_n\}\) have the following properties:
(i) For any positive integer \( k \), \( a_{2k-1} + a_{2k} = 2^k \);
(ii) For any positive integer \( m \), there exists a positive integer \( l \leq m \) such that \( a_{m+1} = \sum_{i=l}^{m} a_i \). Find the maximum value of \( a_2 + a_4 + a_6 + \cdots + a_{2022} \).

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