
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 positive constant  $c_0$  such that the series \[ \displaystyle\sum_{n = 0}^{\infty} \dfrac {n!}{(cn)^n} \] converges for  $c>c_0$  and diverges for  $0<c<c_0$ .

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