
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 that $1$, $a\_1$, $a\_2$, $4$ form an arithmetic sequence, and $1$, $b\_1$, $b\_2$, $b\_3$, $4$ form a geometric sequence, determine the value of $\frac{a\_2 - a\_1}{b\_2}$.

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