
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 the vectors $a=(2,6)$ and $b=(-1, \lambda)$, if $a$ is parallel to $b$, find the value of $\lambda$.

(2) In triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively. Given that $C=60^{\circ}$, $b=\sqrt{6}$, and $c=3$, find the measure of angle $A$.

(3) If the line $\frac{x}{a}+\frac{y}{b}=1 (a>0, b>0)$ passes through the point $(1,2)$, find the minimum value of $2a+b$.

(4) Given the sequence ${{a_n}}$, where ${a_1}=1$ and the sum of the first $n$ terms ${S_n}$ satisfies the equation ${S_n}\sqrt{{S_{n-1}}}-{S_{n-1}}\sqrt{{S_n}}=2\sqrt{{S_n}{S_{n-1}}}$, find ${a_n}$.

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