
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 that the plane vectors \( \boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c} \) satisfy the conditions
\[ |\boldsymbol{a}| = |\boldsymbol{b}| = \boldsymbol{a} \cdot \boldsymbol{b} = \boldsymbol{c} \cdot (\boldsymbol{a} + 2\boldsymbol{b} - 2\boldsymbol{c}) = 2, \]

if the maximum and minimum values of \( |\boldsymbol{a} - \boldsymbol{c}| \) are \( M \) and \( m \) respectively, then find \( \frac{M}{m} \).

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