
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
Given two non-zero vectors \( \vec{a} \) and \( \vec{b} \) in the plane that are perpendicular to each other and \( |\vec{b}|=1 \), find the sum of all real numbers \( m \) such that the vectors \( \vec{a} + m \vec{b} \) and \( \vec{a} + (1-m) \vec{b} \) are perpendicular to each other.

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