
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 \( |\boldsymbol{a}| = 1 \), \( |\boldsymbol{b}| = |\boldsymbol{c}| = 2 \), \( \boldsymbol{b} \cdot \boldsymbol{c} = 0 \), \( \lambda \in (0, 1) \), find the minimum value of 
\[ 
|a - b + \lambda(b - c)| + \left| \frac{1}{2}c + (1 - \lambda)(b - c) \right| 
\]

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