
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 vectors $\overrightarrow{p}=(1,2)$, $\overrightarrow{q}=(m,1)$, if $\vec{p} \bot \vec{q}$, then $|\vec{q}|=$_______.

(2) Teachers A, B, and C work in three different middle schools in Beijing, Shanghai, and Guangzhou, respectively, teaching different subjects A, B, and C. It is known that:

(i) Teacher A does not work in Beijing, and Teacher B does not work in Shanghai; (ii) The teacher working in Beijing does not teach subject C;

(iii) The teacher working in Shanghai teaches subject A; (iv) Teacher B does not teach subject B.

It can be determined that the subject taught by Teacher B is _______.

(3) If the function $f(x)=\begin{cases}2^{x}-3, & x\leqslant 0 \\ x-2, & x > 0\end{cases}$ and $f(a)=1$, then $a=$_______.

(4) Given the sequence $\{a_n\}$ satisfies $a_1=1, a_2=2, a_n=a_{n-2}+2^{n-1}, n\geqslant 3$, then $a_{2016}+a_{2017}=$_________.

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