
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 real numbers $x_{1}$, $x_{2}$, $y_{1}$, $y_{2}$ satisfy: $x_{1}^{2}+y_{1}^{2}=1$, $x_{2}^{2}+y_{2}^{2}=1$, $x_{1}x_{2}+y_{1}y_{2}=\frac{1}{2}$, then the maximum value of $\frac{|{x}_{1}+{y}_{1}-1|}{\sqrt{2}}+\frac{|{x}_{2}+{y}_{2}-1|}{\sqrt{2}}$ is ____.

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