
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 smallest positive real number  $\lambda$  such that for every numbers  $a_1,a_2,a_3 \in \left[0, \frac{1}{2} \right]$  and  $b_1,b_2,b_3 \in (0, \infty)$  with  $\sum\limits_{i=1}^3a_i=\sum\limits_{i=1}^3b_i=1,$  we have   $$ b_1b_2b_3 \le \lambda (a_1b_1+a_2b_2+a_3b_3). $$  

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