
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
For each positive integer \( k \), let \( t(k) \) be the largest odd divisor of \( k \). Determine all positive integers \( a \) for which there exists a positive integer \( n \) such that all the differences
\[ 
t(n+a)-t(n), \quad t(n+a+1)-t(n+1), \quad \ldots, \quad t(n+2a-1)-t(n+a-1) 
\]
are divisible by 4.

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