
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  $\sigma (n)$  denote the sum and  $\tau (n)$  denote the amount of natural divisors of number  $n$  (including  $1$  and  $n$ ). Find the greatest real number  $a$  such that for all  $n>1$  the following inequality is true:   $$ \frac{\sigma (n)}{\tau (n)}\geq a\sqrt{n} $$  

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