
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
We know from problem 966 that if the square-based pyramid $ABCDE$ has edge $AB$ with midpoint $M_2$ and edge $AE$ with midpoint $M_1$, and if we draw plane $S$ through these midpoints as well as through point $M_3$ dividing edge $BC$ in the ratio $1:2$, this plane intersects the other edges of the pyramid $CE$ and $DE$ at points $M_4$ and $M_5$ in the ratios $2:1$ and $4:1$ respectively.

Moreover, the extensions of the base edges $DA$ and $DC$, and the diagonal $DB$ are intersected by the plane $S$ at points $N_1$, $N_2$, and $N_3$ such that $DN_1=4AN_1$, $DN_2=2CN_2$, and $DN_3:N_3B=4:1$. We also know that plane $S$ is parallel to the edge $BE$.

Calculate the volumes of the two solids formed by this intersection if the height of the original pyramid is equal to the length of a base edge. (It is advisable to first calculate the volume of the pyramid $M_5AM_2M_3CD$.)

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