
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 \(ABC D - A_1 B_1 C_1 D_1\) with an edge length of 1, a plane passes through vertex \(C_1\) and intersects the midpoint of \(A_1 D_1\) at point \(E\), a point \(F\) on \(A_1 A\) that is closer to \(A\) and divides it into three equal segments, point \(G\) on \(A B\), and point \(H\) on \(B C\). Find the perimeter of the cross-sectional hexagon \(C_{1} E F G H\).

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