
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 the vectors \(\boldsymbol{a} = (x, 1)\), \(\boldsymbol{b} = (2, y)\), and \(\boldsymbol{c} = (1,1)\), and knowing that \(\boldsymbol{a} - \boldsymbol{b}\) is collinear with \(\boldsymbol{c}\). Find the minimum value of \( |\boldsymbol{a}| + 2|\boldsymbol{b}| \).

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