
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 $\{a\_n\}$ is an arithmetic sequence with a non-zero common difference, and $\{b\_n\}$ is a geometric sequence such that $a\_1=3$, $b\_1=1$, $a\_2=b\_2$, $3a\_5=b\_3$. If $a\_n=a\_1+\log \_ab\_n$ holds true for every positive integer $n$, find the constant $a=\_\_\_\_\_\_\_\_$.

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