
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 $\varphi(n)$ denote the count of natural numbers less than $n$ that are coprime to $n$, and let $\sigma(n)$ denote the sum of the divisors of $n$. Determine all exponents $k$ for which

$$
\varphi\left(\sigma\left(2^{k}\right)\right)=2^{k}
$$

holds true. (To assist, we reveal that 641 is a divisor of $\left(2^{32}+1\right)$.)

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