
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
Each pair  $(x, y)$  of nonnegative integers is assigned number  $f(x, y)$  according the conditions: $f(0, 0) = 0$ ; $f(2x, 2y) = f(2x + 1, 2y + 1) = f(x, y)$ , $f(2x + 1, 2y) = f(2x, 2y + 1) = f(x ,y) + 1$  for  $x, y \ge 0$ .
Let  $n$  be a fixed nonnegative integer and let  $a$ ,  $b$  be nonnegative integers such that  $f(a, b) = n$ . Decide how many numbers satisfy the equation  $f(a, x) + f(b, x) = n$ .

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