
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 the plane vectors $\vec{a} = \left(2^{m}, -1\right)$ and $\vec{b} = \left(2^{m} - 1, 2^{m+1}\right)$ are perpendicular, where $m$ is a real number, then the magnitude of $\vec{a}$ is .

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