
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 every positive integer  $k$  let  $a(k)$  be the largest integer such that  $2^{a(k)}$  divides  $k$ . For every positive integer  $n$  determine  $a(1)+a(2)+\cdots+a(2^n)$ .

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