
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 $min\{ a,b\}$ represents the smaller value between the real numbers $a$ and $b$, and the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ satisfy $(\vert\overrightarrow{a}\vert=1,\vert\overrightarrow{b}\vert=2,\overrightarrow{a}\cdot\overrightarrow{b}=0,\overrightarrow{c}=\lambda\overrightarrow{a}+\mu\overrightarrow{b}(\lambda+\mu=1))$, find the maximum value of $min\{\overrightarrow{c}\cdot\overrightarrow{a}, \overrightarrow{c}\cdot\overrightarrow{b}\}$ and the value of $\vert\overrightarrow{c}\vert$.

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