
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
Determine the conditions that the non-negative integers $a_{1}, b_{1}, c_{1}, d_{1}$ must satisfy so that there exists an integer quadruple $a_{0}, b_{0}, c_{0}, d_{0}$ for which the following holds:

$$
\left|a_{0}-b_{0}\right|=a_{1}, \quad\left|b_{0}-c_{0}\right|=b_{1}, \quad\left|c_{0}-d_{0}\right|=c_{1}, \quad\left|d_{0}-a_{0}\right|=d_{1}
$$

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