
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
a) For each  $n \ge 2$ , find the maximum constant  $c_{n}$  such that
 $\frac 1{a_{1}+1}+\frac 1{a_{2}+1}+\ldots+\frac 1{a_{n}+1}\ge c_{n}$ 
for all positive reals  $a_{1},a_{2},\ldots,a_{n}$  such that  $a_{1}a_{2}\cdots a_{n}= 1$ .

b) For each  $n \ge 2$ , find the maximum constant  $d_{n}$  such that
 $\frac 1{2a_{1}+1}+\frac 1{2a_{2}+1}+\ldots+\frac 1{2a_{n}+1}\ge d_{n}$ 
for all positive reals  $a_{1},a_{2},\ldots,a_{n}$  such that  $a_{1}a_{2}\cdots a_{n}= 1$ .

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