
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
Find all pairs of integers  $a, b$  such that the following system of equations has a unique integral solution  $(x , y , z )$  : 
  $\begin{cases}x + y = a - 1 
x(y + 1) - z^2 = b \end{cases}$ 

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