
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 the number of integers  $a$  with  $1\le a\le 2012$  for which there exist nonnegative integers  $x,y,z$  satisfying the equation
\[x^2(x^2+2z) - y^2(y^2+2z)=a.\]

*Ray Li.*

<details><summary>Clarifications</summary>[list=1][*] $x,y,z$  are not necessarily distinct.[/list]</details>

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