
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 largest constant  $c>0$  such that for every positive integer  $n\ge 2$ , there always exist a positive divisor  $d$  of  $n$  such that   $$ d\le \sqrt{n}\hspace{0.5cm} \text{and} \hspace{0.5cm} \tau(d)\ge c\sqrt{\tau(n)} $$   where  $\tau(n)$  is the number of divisors of  $n$ . 

*Proposed by Mohd. Suhaimi Ramly*

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