
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  $100m+n$  if  $m$  and  $n$  are relatively prime positive integers such that \[ \sum_{\substack{i,j \ge 0  i+j \text{ odd}}} \frac{1}{2^i3^j} = \frac{m}{n}. \]*Proposed by Aaron Lin*

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