
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 $ k>0 $, let $ I_k\equiv10\ldots064 $ , where there are $ k $ zeros between the $ 1 $ and the $ 6 $ . Let $ N(k) $ be the number of factors of $ 2 $ in the prime factorization of $ I_k $. Calculate the maximum value of $ N(k) $.

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