
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 that $\cos \left( \alpha + \frac{\pi }{4} \right) = \frac{3\sqrt{2}}{5}$, find the value of $\sin 2\alpha$.
(2) Given vectors $\overrightarrow{a} = \left( m, 3 \right)$ and $\overrightarrow{b} = \left( -2, 2 \right)$, where $\left( \overrightarrow{a} - \overrightarrow{b} \right) \bot \overrightarrow{b}$, find the value of $m$.
(3) Given a sequence ${{a}_{n}}$ such that ${{a}_{m}} + {{a}_{n}} = {{a}_{m+n}}$ for any $m, n \in \mathbb{N}^{+}$ and ${{a}_{1}} = 2$, find the value of ${{a}_{2018}}$.
(4) Given a circle $C: (x - 4)^{2} + (y + 2)^{2} = 5$, a line $y = x + 2$, and a point $M$ on the line that is closest to the circle center, if $MN$ is tangent to the circle $C$ at point $N$, find the length of the line segment $MN$.

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