
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
Determine maximum real  $ k$  such that there exist a set  $ X$  and its subsets  $ Y_{1}$ ,  $ Y_{2}$ ,  $ ...$ ,  $ Y_{31}$  satisfying the following conditions: 
(1) for every two elements of  $ X$  there is an index  $ i$  such that  $ Y_{i}$  contains neither of these elements;

(2) if any non-negative numbers  $ \alpha_{i}$  are assigned to the subsets  $ Y_{i}$  and  $ \alpha_{1}+\dots+\alpha_{31}=1$  then there is an element  $ x\in X$  such that the sum of  $ \alpha_{i}$  corresponding to all the subsets  $ Y_{i}$  that contain  $ x$  is at least  $ k$ .

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