
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
For all integers  $n\geq 1$  we define  $x_{n+1}=x_1^2+x_2^2+\cdots +x_n^2$ , where  $x_1$  is a positive integer. Find the least  $x_1$  such that 2006 divides  $x_{2006}$ .

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