
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 distance from the center of the circle $x^2+y^2-2x-4y=0$ to the line $x-y+a=0$ is $\frac{\sqrt{2}}{2}$, then the value of $a$ is _______.

(2) Given that the geometric sequence $\{a_n\}$ satisfies $a_1=3$, and $4a_1$, $2a_2$, $a_3$ form an arithmetic sequence, then the common ratio of this sequence equals _______.

(3) For vectors $a=(\lambda,2)$ and $b=(-3,5)$, and the angle between $a$ and $b$ is acute, then the range of values for $\lambda$ is _______.

(4) Consider the following statements:

① For lines $l_1, l_2$ with slope angles $\alpha_1, \alpha_2$ respectively, if $l_1 \bot l_2$, then $|\alpha_1-\alpha_2|=90^\circ$;

② For any angle $\theta$, the angle between vectors $e_1=(\cos \theta,\sin \theta)$ and $e_2=(\cos \theta- \sqrt{3}\sin \theta, \sqrt{3}\cos \theta+\sin \theta)$ is $\frac{\pi}{3}$;

③ If $\Delta ABC$ satisfies $\frac{a}{\cos B}=\frac{b}{\cos A}$, then $\Delta ABC$ is necessarily an isosceles triangle;

④ For any positive numbers $a,b$, it holds that $1 < \frac{\sqrt{a}+\sqrt{b}}{\sqrt{a+b}}\leqslant \sqrt{2}$.

The correct statement(s) is/are _______.

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