
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

A player on the computer controls a fox hunting two hares. There is a den at vertex \( A \) of the square \( ABCD \); if even one hare reaches it in the fox's absence, the game is lost. The fox catches a hare as soon as it finds itself at the same point as the hare (potentially at point \( A \)). Initially, the fox sits at point \( C \) and the hares at points \( B \) and \( D \). The fox can move anywhere with a speed not exceeding \( v \), while the hares move along the rays \( AB \) and \( AD \) with a speed not exceeding 1. For which values of \( v \) can the fox catch both hares?

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