
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
Given two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$, with magnitudes $|\overrightarrow{a}|=1$ and $|\overrightarrow{b}|=2$, and $(\overrightarrow{a} + \overrightarrow{b}) \perp \overrightarrow{a}$. In triangle $ABC$, $\overrightarrow{AB} = \overrightarrow{a}$ and $\overrightarrow{AC} = \overrightarrow{b}$. Let $D$ be the midpoint of $BC$. Determine the length of $AD$.
A) $\frac{\sqrt{7}}{2}$
B) $\frac{\sqrt{6}}{2}$
C) $\frac{\sqrt{5}}{2}$
D) $\frac{\sqrt{3}}{2}$

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