
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
A sequence $\{a_n\}$ with the sum of its first $n$ terms denoted by $S_n$ is called a harmonic series if $\frac{S_n}{S_{2n}}$ is a constant. If an arithmetic sequence with first term 1 and common difference $d$ (where $d \neq 0$) is a harmonic series, then the common difference $d$ equals _____________.

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