
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
In the triangular prism $ABC-A_{1}B_{1}C_{1}$, the side edge $A_{1}A$ is perpendicular to the base $ABC$, $AB=2$, $AC=1$, $AA_{1}=2$, $\angle BAC=90^{\circ}$. The cosine of the angle between line $AB_{1}$ and line $A_{1}C$ is ______.

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