
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
The sum of the first three terms of an arithmetic sequence, as well as the sum of the first seven terms of the same sequence, are natural numbers. Additionally, the first term \( c_{1} \) of the sequence satisfies the inequality \( c_{1} \geqslant \frac{1}{3} \). What is the smallest possible value of \( c_{1} \)?

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