
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
Given an integer  $n \ge 2$ , determine the number of ordered  $n$ -tuples of integers  $(a_1, a_2,...,a_n)$  such that
(a)  $a_1 + a_2 + .. + a_n \ge n^2$   and
(b)  $a_1^2 + a_2^2 + ... + a_n^2 \le n^3 + 1$ 

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