
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
Find the largest positive integer  $k$  such that  $\phi ( \sigma ( 2^k)) = 2^k$ . ( $\phi(n)$  denotes the number of positive integers that are smaller than  $n$  and relatively prime to  $n$ , and  $\sigma(n)$  denotes the sum of divisors of  $n$ ). As a hint, you are given that  $641|2^{32}+1$ .

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