
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 between numeral systems; convert from base 5 to base 7: $412_{(5)} = \_\_\_\_\_$$_{(7)}$.

(2) Given proposition $p$: $\exists x_{0} \in \mathbb{R}$, $ax_{0}^{2}+2x_{0}+a < 0$, and proposition $q$: The equation $x^{2}+ax+1=0$ has two distinct negative roots. If $p \lor q$ is true, then the range of real number $a$ is $\_\_\_\_\_$.

(3) If the equation $\frac{x^{2}}{k-2}+ \frac{y^{2}}{3-k}=1$ represents an ellipse, then the range of $k$ is $\_\_\_\_\_$.

(4) Given the space vectors $\overrightarrow{a} =(1,2,3)$, $\overrightarrow{b} =(1,-1,1)$, the vector set $A=\{ \overrightarrow{x} | \overrightarrow{x} = \overrightarrow{a} +k \overrightarrow{b} ,k\in\mathbb{Z}\}$, then the minimum value of the magnitude of vector $\overrightarrow{x}$, $|\overrightarrow{x}|$, is $\_\_\_\_\_$.

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