
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 terms of a sequence are:

$$
\begin{aligned}
&a_{0}=1, \quad a_{1}=1-\left(\frac{1}{2}\right)^{2} a_{0}, \quad a_{2}=1-\left(\frac{2}{3}\right)^{2} a_{1} \\
&a_{3}=1-\left(\frac{3}{4}\right)^{2} a_{2}, \ldots, \quad a_{n}=1-\left(\frac{n}{n+1}\right)^{2} a_{n-1}
\end{aligned}
$$

Express \(a_{n}\) using a formula that does not involve any previous term in the sequence. Also, write a formula for the following product:

$$
a_{0} a_{1} a_{2} a_{3} \ldots a_{n}
$$

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