
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
On Day  $1$ , Alice starts with the number  $a_1=5$ . For all positive integers  $n>1$ , on Day  $n$ , Alice randomly selects a positive integer  $a_n$  between  $a_{n-1}$  and  $2a_{n-1}$ , inclusive. Given that the probability that all of  $a_2,a_3,\ldots,a_7$  are odd can be written as  $\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers, compute  $m+n$ .


*2021 CCA Math Bonanza Lightning Round #1.4*

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