
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
For  $i=1,2,\ldots,7$ , Zadam Heng chooses a positive integer  $m_i$  at random such that each positive integer  $k$  is chosen with probability  $\frac{2^i-1}{2^{ik}}$ . If  $m_1+2m_2+\ldots+7m_7\neq35$ , Zadam keeps rechoosing the  $m_i$  until this equality holds. Given that he eventually stops, what is the probability that  $m_4=1$  when Zadam stops?

*2019 CCA Math Bonanza Tiebreaker Round #3*

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