
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
Let the sequence $\left\{ a_n \right\}$ satisfy $a_1 + 3a_2 + \ldots + (2n-1)a_n = 2n$.

$(1)$ Find the general formula for the sequence $\left\{ a_n \right\}$;

$(2)$ Find the sum of the first $n$ terms of the sequence $\left\{ \frac{a_n}{2n+1} \right\}$;

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