
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 {${a_n}$} is an arithmetic sequence, ${a_}_{1}+{a_}_{3}+{a_}_{5}=105$, ${a_}_{2}+{a_}_{4}+{a_}_{6}=99$, find ${a_}_{20}$.

(2) Given that $|a|=3$, $|b|=5$, and the projection of vector $a$ in the direction of vector $b$ is $\frac{12}{5}$, find $a \cdot b$.

(3) In $\triangle ABC$, given that $|\overrightarrow{AB}|=|\overrightarrow{AC}|=4$, and $\overrightarrow{AB} \cdot \overrightarrow{AC}=8$, find the shape of this triangle.

(4) Given that forces ${F_}_{1}$, ${F_}_{2}$, ${F_}_{3}$ satisfy $|{F_}_{1}|=|{F_}_{2}|=|{F_}_{3}|=1$, and ${F_}_{1}+{F_}_{2}+{F_}_{3}=0$, find $|{F_}_{1}-{F_}_{2}|$.

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