
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
Let \(\vec{a}\) and \(\vec{b}\) be two perpendicular plane vectors, and \( |\vec{a}| = 2|\vec{b}| = 10 \). For \( 0 \leqslant t \leqslant 1 \), let the vectors \(t \vec{a}+(1-t) \vec{b}\) and \(\left(t-\frac{1}{5}\right) \vec{a}+(1-t) \vec{b}\) have a maximum angle \(\theta\). Then, \(\cos \theta =\) ____.

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