
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) Convert the ternary number $121_{(3)}$ to a decimal number.

(2) If the proposition "$\exists x \in \mathbb{R}$, such that $x^{2}+(a-1)x+1 < 0$" is false, then the range of the real number $a$ is.

(3) Given that the derivative of the function $f(x)$ is $f′(x)$, and it satisfies $f(x)=3x^{2}+2xf′(2)$, then $f′(5)=$.

(4) Among the following five propositions about conic sections, the numbers of the true propositions are (write down all the numbers of the true propositions):

① The hyperbola $\dfrac{{x}^{2}}{16}- \dfrac{{y}^{2}}{9} =1$ and the ellipse $\dfrac{{x}^{2}}{49}+ \dfrac{{y}^{2}}{24} =1$ have the same foci.

② The circle with the diameter being the focal chord of the parabola (the line segment cut by a line passing through the focus of the parabola) is tangent to the directrix of the parabola.

③ If $A$ and $B$ are two fixed points, and $k$ is a constant, if $|PA|-|PB|=k$, then the locus of point $P$ is a hyperbola.

④ For a given circle $C$ and a point $A$ on it, if a moving chord $AB$ is drawn, and $O$ is the origin, if $\overset{→}{OP}= \dfrac{1}{2}\left( \overset{→}{OA}+ \overset{→}{OB}\right)$, then the locus of point $P$ is an ellipse.

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