
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
(1) Given that the angle between unit vectors $e_1$ and $e_2$ is $30^{\circ}$, then $|e_1 - \sqrt{3}e_2| = \_\_\_\_\_\_\_\_$.

(2) In an acute triangle $ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, if $(a^2 + b^2 - c^2) \tan C = ab$, then the value of angle $C$ is $\_\_\_\_\_\_\_\_$.

(3) Let $_{S_n}$ be the sum of the first $n$ terms of a positive geometric sequence $\{a_n\}$, and $S_4 = 5S_2$, then the value of $\frac{a_3 \cdot a_8}{a_5^2} = \_\_\_\_\_\_\_\_$.

(4) Given $\alpha$ and $\beta$ are two different planes, and $m$ and $n$ are two different lines, the following conclusions are drawn:

① If $m$ and $n$ are parallel to the same plane, then $m$ and $n$ are parallel;

② If $m \perp \alpha$ and $n \parallel \alpha$ then $m \perp n$;

③ If $\alpha$ and $\beta$ are not parallel, then there is no line in $\alpha$ that is parallel to $\beta$;

④ If $\alpha \bigcap \beta = n$ and $m \parallel n$, then $m \parallel \alpha$ and $m \parallel \beta$;

⑤ If $m \parallel n$ and $\alpha \parallel \beta$, then the angle formed by $m$ with $\alpha$ is equal to the angle formed by $n$ with $\beta$.

Among these, the correct ones are $\_\_\_\_\_\_\_\_$. (Fill in the numbers of all correct conclusions)

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