
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 \(x\), \(y\), \(z\), and \(r\) be real numbers. Find the minimum value of the following algebraic expressions:
1. \(x^{2}+y^{2}+xy+x+y\)
2. \(x^{2}+y^{2}+z^{2}+xy+yz+zx+x+y+z\)
3. \(x^{2}+y^{2}+z^{2}+r^{2}+xy+xz+xr+yz+yr+zr+x+y+z+r\)

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