
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 $\overrightarrow{a}=(1,1)$ and $\overrightarrow{b}=(4,x)$, if $\overrightarrow{c}=\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{d}=2\overrightarrow{a}+\overrightarrow{b}$, and $\overrightarrow{c}$ is parallel to $\overrightarrow{d}$, then $x=$_______.

(2) Given that $\alpha \in \left(0,\frac{\pi}{4}\right)$ and $\sin \alpha +\cos \alpha =\frac{3\sqrt{2}}{4}$, then $\cos \left(\alpha +\frac{\pi}{4}\right)=$__________.

(3) In parallelogram $ABCD$, points $E$ and $F$ are the midpoints of sides $CD$ and $BC$, respectively. If $\overrightarrow{AC}=m\overrightarrow{AE}+n\overrightarrow{AF}$ ($m,n\in \mathbb{R}$), then $m+n=$_______.

(4) In $\triangle ABC$, it is known that $\tan \frac{A+B}{2}=\sin C$. Consider the following conclusions:

$①$ $\triangle ABC$ is a right triangle; $②\frac{\tan A}{\tan B}=1$;  $③ 1 < \sin A+\sin B\leqslant \sqrt{2}$;

$④\sin^2 A+\cos^2 B=1$; $⑤\cos^2 A+\cos^2 B=\sin^2 C$.

The sequence numbers of the correct conclusions are__________. (Write down all correct conclusion numbers).

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