
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) The equation of the line that passes through the origin and is tangent to the curve $y=\ln x$ is _______________.

(2) If the chord cut by the directrix of the parabola $y^{2}=2px (p > 0)$ from the circle $x^{2}+y^{2}-2y-1=0$ has a length of $2$, then $p=$ _______________.

(3) If there exists a positive number $x$ such that $2^{x}+a > 4^{x}$ holds, then the range of the real number $a$ is _______________.

(4) Given a sequence $\{a_{n}\}$ satisfying $a_{1}=0$, $a_{2}=1$, $a_{n+2}=3a_{n+1}-2a_{n}$, then the sum of the first $n$ terms of $\{a_{n}\}$, $S_{n}=$ _______________.

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