
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 vectors $ \vec{a}, \vec{b} $ satisfy $ \vec{a} \cdot \vec{b} = 10 $ and $ |\vec{a} + \vec{b}| = 5\sqrt{2} $, and $ \vec{a} = (2,1) $, calculate the magnitude of vector $ \vec{b} $.

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