
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
Given the arithmetic sequence $\_{a\_n\_}$, where the common difference $d \neq 0$, and the terms $a\_1$, $a\_3$, $a\_4$ form a geometric sequence. Calculate the value of $\frac{a\_1 + a\_5 + a\_17}{a\_2 + a\_6 + a\_18}$.

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