
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 that $min\\{a,b\\}= \\begin{cases} a, & a\\leqslant b \\\\ b, & a > b\\end{cases}$, and the vectors $ \\overrightarrow{a}, \\overrightarrow{b}, \\overrightarrow{c}$ satisfy $| \\overrightarrow{a}|=1,| \\overrightarrow{b}|=2$, the angle between $ \\overrightarrow{a}$ and $ \\overrightarrow{b}$ is $120^{\\circ}$, and $ \\overrightarrow{c}=λ \\overrightarrow{a}+μ \\overrightarrow{b}\\;,λ+μ=2$, find the maximum value of $min\\{ \\overrightarrow{c}\\cdot \\overrightarrow{a}, \\overrightarrow{c}\\cdot \\overrightarrow{b}\\}$, and the value of $| \\overrightarrow{c}|$ when it reaches the maximum.

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