
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 max{p, q} be defined as $$\begin{cases} p, & \text{if } p \geq q \\ q, & \text{if } p < q \end{cases}$$, and let $M(x, y) = \max\{|x^2 + y + 1|, |y^2 - x + 1|\}$, where $x, y \in \mathbb{R}$. Find the minimum value of $M(x, y)$.

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