
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 vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ satisfy $|\overrightarrow{a}|=|\overrightarrow{b}|=2$, and $(\overrightarrow{a}+2\overrightarrow{b})\cdot(\overrightarrow{a}-\overrightarrow{b})=-2$, the angle between vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ is \_\_\_\_\_\_.

(2) Given the function $f(x)=\ln x-3x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(1,-3)$ is \_\_\_\_\_\_.

(3) During the school's art festival, four projects $A$, $B$, $C$, and $D$ participated in the same category, with only one project receiving the first prize. Before the award ceremony, four students, Jia, Yi, Bing, and Ding, made the following predictions:
   Jia said, "Project $C$ or $D$ will receive the first prize";
   Yi said, "Project $B$ will receive the first prize";
   Bing said, "Projects $A$ and $D$ will not receive the first prize";
   Ding said, "Project $C$ will receive the first prize".
   If only two of these students' predictions are correct, the project that received the first prize is \_\_\_\_\_\_.

(4) Given that $x > 0$, $y > 0$ and $x+y=1$, the minimum value of $\frac{1}{x}+\frac{4}{y}$ is \_\_\_\_\_\_.

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