
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 plane vectors $\overrightarrow{a}$, $\overrightarrow{b}$, $\overrightarrow{c}$, satisfying $|\overrightarrow{b}|=2$, $|\overrightarrow{a}+\overrightarrow{b}|=1$, $\overrightarrow{c}=λ\overrightarrow{a}+μ\overrightarrow{b}$ and $\lambda +2\mu =1$. If for each determined vector $\overrightarrow{a}$, the minimum value of $|\overrightarrow{c}|$ is denoted as $m$, then as $\overrightarrow{a}$ varies, the maximum value of the real number $m$ is ______.

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