
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  $n_0$  be the product of the first  $25$  primes. Now, choose a random divisor  $n_1$  of  $n_0$ , where a choice  $n_1$  is taken with probability proportional to  $\phi(n_1)$ . ( $\phi(m)$  is the number of integers less than  $m$  which are relatively prime to  $m$ .) Given this  $n_1$ , we let  $n_2$  be a random divisor of  $n_1$ , again chosen with probability proportional to  $\phi(n_2)$ . Compute the probability that  $n_2\equiv0\pmod{2310}$ .

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