
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 plane vectors \( \boldsymbol{a}, \boldsymbol{b}, \boldsymbol{c} \) satisfying \( |\boldsymbol{a}| = |\boldsymbol{b}| = 1 \) and \( |\boldsymbol{c}| = 5 \), with \( \boldsymbol{a} \cdot \boldsymbol{c} = 3 \) and \( \boldsymbol{b} \cdot \boldsymbol{c} = 4 \), find the minimum value of \( z = |\boldsymbol{c} - t\boldsymbol{a} - \boldsymbol{b}| \) for any real number \( t \).

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