
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) If the function $f(x)=\log_{8}x+\log_{2}x^{2}$, then $f(8)=$_______.

(2) In the expansion of $(x+a)^{9}$, if the coefficient of the fourth term is $84$, then $a=$_______.

(3) Line $l$ passes through the focus $F$ of the parabola $y^{2}=4x$ and intersects the parabola at points $A$ and $B$. If $\overrightarrow{AF}=5\overrightarrow{FB}$, then the slope of line $l$ is _______.

(4) In the sequence $\{a_{n}\}$, $a_{1}=12$, and $\dfrac{a_{n+1}}{3n+4}=\dfrac{3a_{n}}{3n+1}$. Let $S_{n}=\sum\limits_{i=1}^{n}{\dfrac{a_{i}}{3i+1}}$ and $T_{n}=\sum\limits_{i=1}^{n}{\dfrac{a_{i}}{3^{i}}}$, then the correct judgment(s) is(are) _______. (Fill in all correct conclusions' numbers)

① The sequence $\{\dfrac{a_{n}}{3n+1}\}$ is a geometric sequence; ② There exists a positive integer $n$ such that $a_{n}$ can be divided by $11$;

③ $S_{10} > T_{243}$. ④ $T_{21}$ can be divided by $51$.

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