
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
If $\frac{2^{12} + 1}{2^{4} + 1} = 2^{a_{1}} + 2^{a_{2}} + \ldots + 2^{a_{k}}$ where $a_{1}, a_{2}, \ldots, a_{k}$ are natural numbers and $a_{1} < a_{2} < \cdots < a_{k}$, then $k =$ ______

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