
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
Derive the operational rules for the dot product of vectors by analogy with the multiplicative rules for algebraic expressions:
\\(①\\) By analogy with "\\(mn = nm\\)", derive "\\( \overrightarrow{a} \cdot \overrightarrow{b} = \overrightarrow{b} \cdot \overrightarrow{a}\\)";
\\(②\\) By analogy with "\\((m + n)t = mt + nt\\)", derive "\\(( \overrightarrow{a} + \overrightarrow{b}) \cdot \overrightarrow{c} = \overrightarrow{a} \cdot \overrightarrow{c} + \overrightarrow{b} \cdot \overrightarrow{c}\\)";
\\(③\\) By analogy with "\\(t \neq 0, mt = nt \Rightarrow m = n\\)", derive "\\( \overrightarrow{c} \neq 0, \overrightarrow{a} \cdot \overrightarrow{c} = \overrightarrow{b} \cdot \overrightarrow{c} \Rightarrow \overrightarrow{a} = \overrightarrow{b}\\)";
\\(④\\) By analogy with "\\(|m \cdot n| = |m| \cdot |n|\\)", derive "\\(| \overrightarrow{a} \cdot \overrightarrow{b}| = | \overrightarrow{a}| \cdot | \overrightarrow{b}|\\)";
\\(⑤\\) By analogy with "\\((m \cdot n)t = m(n \cdot t)\\)", derive "\\(( \overrightarrow{a} \cdot \overrightarrow{b}) \cdot \overrightarrow{c} = \overrightarrow{a}( \overrightarrow{b} \cdot \overrightarrow{c})\\)";
\\(⑥\\) By analogy with "\\( \frac{ac}{bc} = \frac{a}{b}\\)", derive "\\( \frac{ \overrightarrow{a} \cdot \overrightarrow{c}}{ \overrightarrow{b} \cdot \overrightarrow{c}} = \frac{ \overrightarrow{a}}{ \overrightarrow{b}}\\)". Among the above formulas, the conclusion obtained by analogy that is correct is \_\_\_\_\_\_.

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