
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 in the plane, $a$ and $b$, satisfying $|a|=1$, $|b|=2$, and the angle between $a$ and $b$ is $\frac{\pi}{3}$. If a parallelogram is formed with $a$ and $b$ as adjacent sides, then the length of the shorter one of the two diagonals of this parallelogram is __________.

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