
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 $\vec{a}$, $\vec{b}$, $\vec{c}$ be real vectors satisfying $|\vec{a}|=1$, $|\vec{b}|=2$, $\vec{a} \cdot \vec{b} = 0$, and $\vec{c} = \lambda \vec{a} + \mu \vec{b}$ ($\lambda + \mu = 1$). Find the value of $|\vec{c}|$ when $min\{\vec{c} \cdot \vec{a}, \vec{c} \cdot \vec{b}\}$ attains its maximum value.

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