
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 functions  $f\colon \mathbb{Z}^2 \to [0, 1]$  such that for any integers  $x$  and  $y$ ,
\[f(x, y) = \frac{f(x - 1, y) + f(x, y - 1)}{2}.\]

*Proposed by Yang Liu and Michael Kural*

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