
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
During a trip, I saw Romanesque cathedrals, triumphal arches, waterfalls, and medieval castles. I photographed half of these tourist attractions. I saw three times as many cathedrals as triumphal arches, and as many medieval castles as waterfalls. A quarter of my photographs depict medieval castles. However, I only photographed every second castle; all the triumphal arches I saw are in my photographs. What proportion of the photographs I took feature triumphal arches?

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