
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
Let $D(k)$ denote the number of positive divisors of $k$. For a quadruple of positive integers $(a, b, c, d)$, if
$$
b = a^{2} + 1, \quad c = b^{2} + 1, \quad d = c^{2} + 1,
$$
and $D(a) + D(b) + D(c) + D(d)$ is odd, then $(a, b, c, d)$ is called "green." How many green quadruples $(a, b, c, d)$ are there with $a, b, c, d$ less than 1,000,000?

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