
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 a cube \( ABCDA_1B_1C_1D_1 \), segment \(\overline{B_1K}\) is equal to \(\overline{KC_1}\); \(\overline{CP} = 0.25 \overline{CD}\); \(\overline{C_1M} = 0.25 \overline{C_1D_1}\). Point \(X\) lies on line \(BD_1\). Point \(Y\) is the intersection of lines \(KX\) and \(MP\). Construct point \(X\).

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