
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
The base of an inclined parallelepiped \( ABCDA_1B_1C_1D_1 \) is a rhombus \( ABCD \) with side length \( a \) and an acute angle of \( 60^{\circ} \). The edge \( AA_1 \) is also equal to \( a \) and forms angles of \( 45^{\circ} \) with edges \( AB \) and \( AD \). Find the volume of the parallelepiped.

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