
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 $\{a_n\}$ satisfy $a_1 + \frac{1}{3}a_2 + \frac{1}{5}a_3 + \cdots + \frac{1}{2n-1}a_n = n$.

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

$(2)$ Calculate the sum of the first $60$ terms of the sequence $\left\{ \frac{2}{\sqrt{a_{n+1}} + \sqrt{a_n}} \right\}$, denoted as $T_{60}$.

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