
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
We call  $ A_1, A_2, \ldots, A_n$  an  $ n$ -division of  $ A$  if

(i)  $ A_1 \cap A_2 \cap \cdots \cap A_n \equal{} A$ ,
(ii)  $ A_i \cap A_j \neq \emptyset$ .

Find the smallest positive integer  $ m$  such that for any  $ 14$ -division  $ A_1, A_2, \ldots, A_{14}$  of  $ A \equal{} \{1, 2, \ldots, m\}$ , there exists a set  $ A_i$  ( $ 1 \leq i \leq 14$ ) such that there are two elements  $ a, b$  of  $ A_i$  such that  $ b < a \leq \frac {4}{3}b$ .

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