
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 non-zero vectors $a$ and $b$ that are not collinear, if $|a|=|b|=1$, and the angle between $a$ and $b$ is $120^{\circ}$, find the value of the real number $x$ that minimizes $|a-xb|$ ($x \in \mathbb{R}$).

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