
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
If \( a_{1}, a_{2}, a_{3}, \ldots, a_{n} \) and \( b_{1}, b_{2}, b_{3}, \ldots, b_{n} \) are any real numbers, then
$$
\left(a_{1} b_{1}+a_{2} b_{2}+\ldots+a_{n} b_{n}\right)^{2} \leq \left(a_{1}^2 + a_{2}^2 + \ldots + a_{n}^2\right) \left(b_{1}^2 + b_{2}^2 + \ldots + b_{n}^2\right).
$$

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