
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
Given real numbers $a_{1}, a_{2}, \cdots, a_{n}$ and $b_{1}, b_{2}, \cdots, b_{n}$, the following inequality holds:
$$
\left(a_{1} b_{1}+a_{2} b_{2}+\cdots+a_{n} b_{n}\right)^{2} \leqslant\left(a_{1}^{2}+a_{2}^{2}+\cdots+a_{n}^{2}\right)\left(b_{1}^{2}+b_{2}^{2}+\cdots+b_{n}^{2}\right),
$$

The equality condition is that the sequences $\left(a_{1}, a_{2}, \cdots, a_{n}\right)$ and $\left(b_{1}, b_{2}, \cdots, b_{n}\right)$ are proportional, i.e., there exist non-zero constants $\lambda$ and $\mu$ such that $\lambda a_{i} = \mu b_{i}$ for $i=1,2, \cdots, n$. If we assume that whenever $a_{i} \neq 0$, $b_{i} \neq 0$, then the equality holds if and only if $\frac{b_{1}}{a_{1}}=\frac{b_{2}}{a_{2}}=\cdots=\frac{b_{n}}{a_{n}}$.

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